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Atomistically Informed Fuel Performance Codes

Permanent Link: http://ufdc.ufl.edu/UFE0022427/00001

Material Information

Title: Atomistically Informed Fuel Performance Codes A Proof of Principle Using Molecular Dynamics and FRAPCON Simulation
Physical Description: 1 online resource (54 p.)
Language: english
Creator: Vega, Daniel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: dynamics, frapcon, molecular
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Current limitations on UO2 fuel lifetime are determined largely from physical degradation of the fuel pellets. In order to better understand the behavior of UO2 fuel, new atomic level approaches are being used to simulate the effects of both temperature and burnup. A proof-of-principle study is presented in which the results of atomic-level simulations of the thermal expansion and thermal conductivity of UO2 are integrated into the Light Water Reactor fuel performance code FRAPCON. The beginning of life (BOL) thermal conductivity profile of a fuel pellet, and the evolution of the pellet expansion over its lifetime are calculated. It is found that (i) modifying FRAPCON in such a way as to accept input from the results obtained through atomistic simulations by the integration of atomistic models into FRAPCON is possible for a number of thermo-mechanical properties, and (ii) the properties determined from atomistic simulations yield predictions in FRAPCON that are in good agreement for the BOL thermal conductivity, but less satisfactory for the pellet thermal expansion, due to phenomena thus far unaccounted for by atomic level simulations. Still, by successfully incorporating these simulations into FRAPCON, a basis for a more complete atomistic model is built, demonstrating the potential for more sophisticated and encompassing first principles models.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Daniel Vega.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Tulenko, James S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022427:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022427/00001

Material Information

Title: Atomistically Informed Fuel Performance Codes A Proof of Principle Using Molecular Dynamics and FRAPCON Simulation
Physical Description: 1 online resource (54 p.)
Language: english
Creator: Vega, Daniel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: dynamics, frapcon, molecular
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Current limitations on UO2 fuel lifetime are determined largely from physical degradation of the fuel pellets. In order to better understand the behavior of UO2 fuel, new atomic level approaches are being used to simulate the effects of both temperature and burnup. A proof-of-principle study is presented in which the results of atomic-level simulations of the thermal expansion and thermal conductivity of UO2 are integrated into the Light Water Reactor fuel performance code FRAPCON. The beginning of life (BOL) thermal conductivity profile of a fuel pellet, and the evolution of the pellet expansion over its lifetime are calculated. It is found that (i) modifying FRAPCON in such a way as to accept input from the results obtained through atomistic simulations by the integration of atomistic models into FRAPCON is possible for a number of thermo-mechanical properties, and (ii) the properties determined from atomistic simulations yield predictions in FRAPCON that are in good agreement for the BOL thermal conductivity, but less satisfactory for the pellet thermal expansion, due to phenomena thus far unaccounted for by atomic level simulations. Still, by successfully incorporating these simulations into FRAPCON, a basis for a more complete atomistic model is built, demonstrating the potential for more sophisticated and encompassing first principles models.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Daniel Vega.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Tulenko, James S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022427:00001


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976ccd2facfaf5936ddfae764f782cb8879d49cc







ATOMISTICALLY INFORMED FUEL PERFORMANCE CODES: A PROOF OF
PRINCIPLE USING MOLECULAR DYNAMICS AND FRAPCON SIMULATION





















By

DANIEL ARMANDO VEGA


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2008


































2008 Daniel Armando Vega


































To Sandwich, my patient cohort









ACKNOWLEDGMENTS

I would like to thank Professor Tulenko, Dr. Phillpot, Dr. Sinnott, Taku Wantanabe, and

the rest of the research team that contributed the work, guidance, criticism, and patience required

to make this research possible. In addition, I extend my appreciation to the U.S. Department of

Energy's Nuclear Engineering Research Initiative (NERI) award selection committee for

acknowledging the merit of this investigation and awarding necessary research funds. Lastly, I

would like to thank the University of Florida's Nuclear and Radiological Engineering and

Materials Science Departments for this joint venture.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IST O F T A B L E S ......... ...................... ...................................................... 7

LIST OF FIGURES ................................. .. ..... ..... ................. .8

LIST OF ABBREVIATIONS AND TERMS......................................................................9

A B S T R A C T ........... ................... ............................................................ 10

CHAPTER

1 IN T R O D U C T IO N ...................................................................................... .. .............. 11

2 FRAPCON-3 FUEL PERFORMANCE CODE ......................................... ...............13

2 .1 F R A P C O N O objectives ............................................... ......................... ...................13
2.2 FRAPCON -3 Solution D escription...................................... .......................... ....... 13
2.3 Fuel Rod Thermal Conductivity Response Calculation.......................................14
2.3.1 Fuel Pellet Thermal Energy Distribution and Conduction Model ..................15
2.3.2 FRAPCON Thermal Conductivity Function.......................... ..................17
2.4 Fuel Rod Thermal Expansion and Mechanical Response..............................20
2.4.1 FRACA S-I Deform ation M odel ............................................ ............... 21
2.4.2 Fuel Thermal Expansion Subroutine...................... ....... ..................22
2.5 FRA PCON Code D escription.......................................... ............... ............... 22
2 .6 F R A P C O N Input F ile ......................................................................... ...................23

3 MOLECULAR DYNAMICS MODELLING.......................................................... 30

3.1 M olecular D ynam ics Sim ulation ........................................ ........................... 30
3.2 The A M /M D M odel R results ....................... ........ ........ ...............................31

4 SIM ULATION M ETHODOLOGY ............................................... ............................ 33

4.1 Thermal Conductivity FRAPCON/AMMD Implementation ....................................33
4.2 Thermal Expansion FRAPCON-AM/MD Implementation............... ...................34

5 R E SU L T S A N D O U TL O O K .................................................................. ........ .................39

5.1 FRAPCON-AM/MD Conductivity Results........................................... ...........39
5.2 FRAPCON-AM/MD Thermal Expansion Results....................... ...............40
5.3 Proposed AM /M D M modifications ........................................ .......................... 41
5.4 Conclusions and Recom m endations ........................................ ........................ 42









APPENDIX

A FTH CON .F SOURCE FILE ............................................................................. 45

B FE X PA N .F SO U R C E FIL E ......................................................................... ....................48

C FRAPCON and FRAPCON-AM/MD Input File......................................... ............... 49

D BOL RADIAL TEMPERATURE DISTRIBUTIONS.........................................................50

D.1 FRAPCON BOL Radial Temperature Profile (node 4/12):.................................... 50
D.2. FRAPCON-AM/MD BOL Radial Temperature Profile (node 4/12): ......................... 51

L IST O F R E F E R E N C E S ............................ ............. ........................... .....................................52

B IO G R A PH IC A L SK E T C H ................................................................................ ...................54










LIST OF TABLES


Table


2-1 The FRAPCON input parameters for the Oconee rod model........................................24


page









LIST OF FIGURES


Figure page

2-1 Simplified FRAPCON solution scheme for one time step .............................................25

2-2 Temperature calculation for 1-D assumption ........................................ ............... 26

2-3 Fuel rod tem perature radial distribution ........................................ ........................ 26

2-4 Mesh point layout .................................. ... .. ... ..... .................. 27

2-5 FRAPCON conductivity functions with Ronchi data.......................................................27

2-6 FRA PCON solution schem atic ................................................ .............................. 28

2-7 FRAPCON calling sequence. (boxed names are modified in AM/MD). ..........................29

3-1 Typical U 0 2 lattice for M D sim ulation .................................................. .....................32

4-1 Thermal conductivity predicted by FRAPCON and AM/MD calculations ....... .......... 36

4-2 M odified FTH C O N .F code excerpt ..................................................................................37

4-3 Temperature-dependent strain for FRACAS-I and AM/MD. (relative to 300K). .............38

5-1 Pellet temperature profile for MATPRO and AM/MD cases at BOL............................43

5-2 Lifetim e surface displaced ent of fuel pellet.................................. ........................ 44

5-3 Conceptual model of fission event and lattice effects ....................................... .......... 44









LIST OF ABBREVIATIONS AND TERMS


BOL Beginning of life: freshly fabricated fuel, zero burnup.

Burnup A pseudo-intensive property describing the amount thermal energy released
per unit mass of nuclear fuel. Typically given in units of megawatt days per
kilogram of Uranium (MWd/kg U)

CRUD Chalk river unidentified deposits: corrosion and wear products (rust
particles, etc.) that become radioactive (i.e., activated) when exposed to
radiation.

EOL End of life: time/burnup level associated with the end of a fuel rod's
lifetime, at which point it is removed from the core and becomes spent fuel

LWR Light water reactor: a thermal nuclear reactor that uses ordinary water (light
water) as both its neutron moderator and coolant. Typically, either
Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs).

MD Molecular dynamics

Phonon A quantized lattice vibration, modeled as a means for heat transfer in a rigid
lattice

Polaron A quasi-particle composed of an electron plus its accompanying polarization
field. At high temperatures, becomes significant in heat transfer

PWR Pressurized Water Reactor: a nuclear reactor that uses a pressure vessel
around the core, keeping it under high pressure to prevent water in the
primary cooling loop from boiling. Most reactor worldwide are PWRs.

IMF Inert matrix fuel









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Master of Science

ATOMISTICALLY INFORMED FUEL PERFORMANCE CODES: A PROOF OF
PRINCIPLE USING MOLECULAR DYNAMICS AND FRAPCON SIMULATION

By

Daniel Armando Vega

August 2008

Chair: James S. Tulenko
Major: Nuclear Engineering Sciences

Current limitations on Uranium dioxide fuel lifetime are determined largely from physical

degradation of the fuel pellets. In order to better understand the behavior of Uranium dioxide

fuel, new atomic level approaches are being used to simulate the effects of both temperature and

burnup. A proof-of-principle study is presented in which the results of atomic-level simulations

of the thermal expansion and thermal conductivity of Uranium dioxide are integrated into the

Light Water Reactor fuel performance code FRAPCON. The beginning of life (BOL) thermal

conductivity profile of a fuel pellet, and the evolution of the pellet expansion over its lifetime are

calculated. It is found that (i) modifying FRAPCON in such a way as to accept input from the

results obtained through atomistic simulations by the integration of atomistic models into

FRAPCON is possible for a number of thermo-mechanical properties, and (ii) the properties

determined from atomistic simulations yield predictions in FRAPCON that are in good

agreement for the BOL thermal conductivity, but less satisfactory for the pellet thermal

expansion, due to phenomena thus far unaccounted for by atomic level simulations. Still, by

successfully incorporating these simulations into FRAPCON, a basis for a more complete

atomistic model is built, demonstrating the potential for more sophisticated and encompassing

first principles models.









CHAPTER 1
INTRODUCTION

Having been in operation for over 35 years, LWRs using uranium dioxide (U02) fuel have

been modeled and studied using a variety of neutronic, thermo-hydraulic, and thermo-

mechanical approaches. Traditionally, the development of such simulations has been guided by

studying the macroscopic phenomena characterized in laboratory experiments and during

commercial reactor operations. A key part of the development and qualification of U02 fuel

systems consists of analysis in fuel performance codes such as FRAPCON.1 FRAPCON-3, the

most recent release, encapsulates the behavior of U02 fuel pellets in terms of specific thermo-

mechanical phenomena and carefully determined and validated empirical relationships. This

approach has been extraordinarily successful in providing quantification of fuel properties.

However, the development of the key empirical relationships is extremely time-consuming and

expensive.

In addition, such empirical relationships are often valid only for discrete cases, creating

simulations with proprietary modules; thus, any approaches that could reduce the time and

expense involved while extending the scope of validity of the models would be extremely

valuable. In this investigation, the first steps in using results from atomistic models, specifically

molecular-dynamics simulations, directly as models into FRAPCON-3, is explored. The first

step in this investigation is to study the nature and structure of FRAPCON-3 to determine the

most prudent sections of source code to modify for atomistic simulation. Secondly, an

appropriate description of the Atomistic/Molecular Dynamics Model (AM/MD) is presented.

Then, a discussion of the coupled simulation methodology is presented, in which the integration

of the FRAPCON and AM/MD is explained. Finally, the results obtained by this coupled

simulation are presented for analysis and recommendations.









It is important to note that while atomic-level simulation can be powerful, the modeling of

every component of a nuclear core is substantial and outside the scope of this investigation; thus

atomic level simulation for this project pertains only to the U02 fuel pellets, keeping the thermal

transport of the gap, cladding, and coolant unchanged. In addition, the fuel rod-specific

macroscopic parameters by which fuel performance is based, is composed of a complex set of

relationships describing geometry-dependent phenomena, material region interfaces, and

composition changes. At this point in the AM/MD model, many of these phenomena have not

yet been described. However, the most fundamental aspects of thermo-mechanical properties

modeling, namely the thermal conductivity and thermal expansion, are appropriate candidates for

rigorous first-principles modeling. By first establishing and validating these radically different

models by comparing them to established models, the avenue for first principles investigations of

yet to be modeled fuel lattices becomes widened. Such a bridge from empirical to first-

principles models is a theme central to this investigation.

This research is funded by DOE-NERI Award DE-FC07-05ID14649, as part of the

overall Global Nuclear Energy Partnership (GNEP) initiative.2









CHAPTER 2
FRAPCON-3 FUEL PERFORMANCE CODE

2.1 FRAPCON Objectives

Fundamentally, the FRAPCON series was created to accurately calculate the performance

of LWR fuel rods3: a major objective of the reactor safety research program sponsored by the

U.S. Nuclear Regulatory Commission (NRC). As a result of an extensive analytical performance

code development program, FRAPCON was created as a joint effort between Idaho National

Engineering and Environmental Laboratory (INEEL) and Pacific Northwest National Laboratory

(PNNL). INEEL has since become Idaho National Laboratory (INL), and both facilities are now

operated by Battelle. With the creation of such a code, the NRC has the ability to not only

achieve the object of accurately calculating thermo-mechanical properties and performance of

fuel rods, but to independently investigate design parameters put forth by both nuclear vendors

and utilities without using proprietary performance codes. By using both in-reactor and out-of-

reactor experiments to benchmark and asses the code capabilities, FRAPCON has proven to be a

reliable tool for simple analysis of LWR fuel rod performance. It is important to note that

FRAPCON is a "steady state" code, which refers to the situation where (i) short-term changes

are sufficiently small that they can be considered constant for a discreet time step, and (ii)

boundary conditions for each iteration do not change. This analysis is valid for constant power

and slow up-ramp or down-ramp operations typical of normal commercial reactor operations.

Reactor accidents and transient analysis are included in FRAPTRAN, and not discussed in this

investigation.

2.2 FRAPCON-3 Solution Description

In order to evaluate steady state fuel rod performance, FRAPCON-3 uses an iterative

process based on defined boundary conditions at each time step. Simply put, the interrelated









effects of fission rate, thermal energy distribution, fuel and cladding temperature, fission product

release, fuel swelling/densification, irradiation-induced growth, thermal expansion, crud

deposition, cladding corrosion are coupled together and run through calculation loops until

temperature solutions converge to within an acceptable error range. Naturally, some of these

phenomena are fuel rod power/temperature specific, while others are cumulative, and heavily

dependent on burnup and previous conditions. For this reason, a collection of loops is used to

calculate the properties for each time step. Because the number and length of time steps is user

defined, it is apparent that increased accuracy can be achieved with finer time steps, while

computational demand can be assuaged by more coarse time steps. Much of the information in

this section is adapted from the FRAPCON manual,3 and the components most pertinent to this

investigation are included in this section. At the beginning of each time step, a number of

operations are carried out:

1. Initial fuel rod data is determined by input from previous step (or initial condition)

2. Temperature of fuel and cladding are computed

3. Fission product generation & release, internal gas pressure computed

4. Temperature response is iterated through loop until solution converges to less than 1% of
AT

When the temperature converges, the parameters are all recorded, sent to the output file,

and used for the initial conditions for the next time step. Figure 2-1 gives a simplified solution

scheme flow chart.

2.3 Fuel Rod Thermal Conductivity Response Calculation

To further simplify heat transport calculations, each calculation for the fuel pellet

temperature distribution is taken in such a way that heat transfer is only in the radial direction.

This is a valid assumption if one assumes (i) azimuthal homogeneity (axissymmetric analysis),









and (ii) for each axial node, heat transfer in the axial direction is negligible as compared to that

in the radial direction, and is thus ignored (Figure 2-2).

The centerline temperature of the pellet, being the innermost and hottest component, is the

first temperature calculated, followed by the pellet surface temperature, the inner cladding

temperature, the outer cladding temperature, the oxidation layer temperature, and finally, the

bulk coolant temperature. In simple terms, the temperature change from the fuel centerline to the

bulk coolant can be given in terms of component temperature changes by the equation:

Tfc (z) = Tbu (z) + ATflm (z) + ATcmd (z) + ATox (z) + ATcad () + ATgap (z) + ATfuel (z) (1-1)
where
Tf = fuel centerline temperature
Tbul = bulk coolant temperature
ATfl = temperature change through the forced convection film layer
ATcmd = temperature change through crud layer
ATox = temperature change through oxide Layer
ATclad = temperature change through cladding
ATgap = temperature change through gap
ATuel = temperature change through fuel pellet

2.3.1 Fuel Pellet Thermal Energy Distribution and Conduction Model

While conduction through each region is important, for the purposes of this section, only

the conduction through the pellet is analyzed. This is because the pellet is the only region to

which fundamental source code changes are made for the FRAPCON-AM/MD investigation.

The finite difference (FD) heat conduction models used on FRAPCON-3 are based on

those presented in RELAP-5.4 Instead of the method of weighted residuals used in previous

releases of FRAPCON, the FD approach used in FRAPCON-3 must contain fine-mesh

capabilities for high burnup analysis, and must interface with other burnup models either in

existence or yet to be created.3









While an FD approach is straightforward for localized or uniform heat sources, the non-

uniformity of internal energy production requires that the FD method take into account the

spatial dependence of internal heat sources. In addition, because FRAPCON-3 contains a user-

defined number of temperature nodes, as well as variable mesh spacing, the FD method must

allow for such flexibility. The steady-state integral form of the heat conduction equation is:


ff k(T, x)VT(x) nis = ffS()a V (1-2)

Where

k = thermal conductivity (W/m-K)

s = surface of control volume (m2)

S = surface normal unit vector

S = internal heat source (W/m3)

T = temperature (K)

V = control volume (m3)

7 = 1-D space coordinate

It is apparent that Equation 1-2 is complex enough in 1-D that thousands of FD

calculations require a fair amount of computation. 2-D and 3-D models of this variety would

require far greater computation resources. Figure 2-4 shows the 1-D mesh point layout. In order

to maintain proper control surface and volume integrals, it is an important assumption that the

geometry remains fixed for this stage in the analysis. In addition, boundary conditions must be

established. For all FD calculations in the fuel pellet, the following boundary conditions are

used:

aT
1) 0T 0 (pellet center is local maximum, azimuthal symetry)


2) Pellet Surface Temperature (Tfs) is defined









Once a suitable temperature distribution is established, FRAPCON-3 can exit the fuel

pellet temperature subroutine and continue with gap, cladding, oxide, crud, and bulk temperature

calculations. When a distribution has been established for the entire rod, the thermal expansion

and stress/strain relationships expressed in other subroutines can be evaluated within the same

coarse time step.

In words, equation 1-2 states that for a steady state fuel pellet, the integrated surface heat

flux through the control area into the gap (left hand side) is equal to the integrated heat

generation rate in the control volume (right hand side). With this in mind, the numerical

solutions to this equation are based on correct balance of temperature and conductivity,

equivalent to a heat generation rate. When incorporating an AM/MD model, there is no reason

to re-evaluate the FD methodology. In fact, in order to discover how the materials properties

affect the overall performance of the fuel rod, it is important to maintain code structure for the

fuel pellet temperature distribution. Returning to equation 1-2, the only parameter that should

change is the thermal conductivity k(T, x) function. In order to understand where this term

comes from, as well as how it compares to an AM/MD model, the following section presents a

description of the thermal conductivity function.

2.3.2 FRAPCON Thermal Conductivity Function

In general, and in the context of FRAPCON, thermal conductivity (k) has a dependence on

temperature. The thermal conductivity function chosen as a base case for use throughout this

investigation is and currently used in FRAPCON-3.3 is one presented by staff at Nuclear Fuel

Industries, Ltd. (Japan), at the May1997 American Nuclear Society Topical meeting on Light

Water Reactor Fuel Performance.5 The NFI function is of the same form as that originally

included in FRAPCON-3 by Lucata et al.6 Equations 1-3 and 1-4 represent these functions,









respectively. While the functions are similar, the NFI function was chosen, because it is

included in the most recent FRAPCON-3 release (FRAPCON-3.3). This function has been

successfully benchmarked, and provides a better fit to collected data from a study by Ronchi.8

The study provides a suitable base for typical PWR fuel conductivity calculations:

1.0 3.50x109 -16361
ko= +( )e T NFI (1-3)
(0.0452+0.000246 x T) T2

1.0 4.715 x 109 -16361
ko= +( )e T Lucata (1-4)
S(0.0375+0.0002165 xT) T2


where

ko = conductivity of unirradiated urania (UO2)

T = temperature (K)

The gadolinia content for this fuel pellet is 0. The conductivity ofunirradiated 95%

dense (theoretical) urania thus becomes the starting point for burnup calculations. Figure 2-57

gives a plot of the NFI function, Lucuta function, and experimental data, collected by Ronchi, et

al, 1999.8 The Ronchi (1999) data come from a study done on unirradiated pellet material at

nominal and high temperatures. A final function describing the conductivity used for calculation

throughout the time step is given by multiplying ko by a collection of four coefficients

representing particular burnup and non-burnup phenomena. These coefficients are:

FD dissolved fission product (temperature and burnup dependent)

FP precipitated fission product (temperature and burnup dependent)

FM Maxwell porosity factor (porosity fraction and shape factor)

FR radiation effect (temperature dependent)









FD becomes an increasingly significant coefficient as temperature and burnup increase, as

is evident in its definition:

(1-5)
FD 1.09 0.0643 T- arctan 1
B3.265 1 1.09 0.0643
B3.265 +

where

B = burnup (atom %) 1 atom% = 9.383 GWd/MTU at 200 MeV/fission

T = temperature (K)

FP also increases with burnup and temperature, and is given by:


0.019B 1 (1-6)
FP = 1 +L20 --
3-0.019B 1200-T
+e 100 )

FM, which is related only to porosity, is given by:

FM = -P (1-7)
l+(s-1)p

where

p = porosity fraction (as fabricated plus swelling)

s = shape factor (1.5 for spheres)

FR, which is temperature-dependent and always applied, is given by:



FR =1- 0.2 (1-8)









Finally, these coefficients are put together and multiplied by ko to give a temperature-

burnup dependent thermal conductivity value (Equation 1-9) and the FD interval of interest is

used for the corresponding temperature, to create a straightforward calculation for instantaneous

thermal conductivity of the fuel pellet:

k = k (FD x FP x FM x FR) (1-9)

This calculation is then carried out for each radial node, after which other FD calculations

are performed for the gap, cladding, oxide, and film layers. The thermal distributions for non-

fuel regions will not be discussed in this investigation.

2.4 Fuel Rod Thermal Expansion and Mechanical Response

Thermal expansion is important to study because its effects lead to structural stress and

corresponding phenomena, as well as changing the thermal conductivity coefficient. In addition,

proper modeling of thermal expansion is necessary to accurately model other geometry-

dependent phenomena, such as fission gas release.

FRAPCON-3 uses the FRACAS-I9 mechanical model to calculate the fuel and cladding

mechanical deformation throughout reactor operation. It is important to note that there are many

components to the overall concept of mechanical response. In fact, FRAPCON uses FRACAS-I

coupled with other non-specific thermal expansion models to develop an encompassing solution

that describes the state of the fuel rod taking into account geometrical considerations. It involves

a complex set of relationships describing geometry-dependent phenomena, region interfaces, and

composition changes. At this point in the AM/MD model, such macroscopic phenomena are

impossible to describe. For this reason, all FRAPCON AM/MD calculations will retain the

geometry-dependent and burnup correction factors of FRAPCON, while the nature of the

theoretical thermal expansion regimes is changed from the empirical FRAPCON model to the









first-principles AM/MD model. From a programming standpoint, this change may seem minor,

but the substitution occurs in the most fundamental and physically significant section of code,

and so therefore has the potential to have a profound effect on the overall performance of the

code. To illustrate this point, Figure 2-63 gives a detailed flow chart of the overall FRAPCON

process, highlighting the specific sections that are to be modified for the FRAPCON AM/MD

model.

2.4.1 FRACAS-I Deformation Model

As mentioned previously, the FRACAS-I model is designed to analyze fuel and cladding

mechanical deformation in order to better simulate the effects of burnup and temperature on the

fuel pin. FRACAS can model two scenarios. The first of these is an "open gap" situation, when

the fuel and cladding are not in contact. This "open gap" situation has cladding with internal and

external pressures that must be calculated. The second scenario is when the fuel has expanded

enough to be in contact with the cladding. In this "closed gap" situation, the fuel drives the

cladding outwards. Overall, the calculations take into account the effects of fuel thermal

expansion, fuel swelling, fuel densification, fuel relocation, cladding thermal expansion,

cladding creep, cladding plasticity, and fission gas/external coolant pressures. Because the gap

closure is a function of burnup, the FRACAS-I model determines gap thickness as a function of

time, and determines the appropriate open-gap or closed-gap model. Fuel expansion leading to a

closed gap depends on both thermal factors and radiation factors; thus, to properly calculate

lifetime strain both the open gap and closed gap models were used for the analysis. At the

current stage of analysis, no changes have been made to the gap conductance or fuel cladding

regimes in the FRACAS-I model. The components of the FRACAS-I model which are not

explicitly used to calculate the theoretical thermal expansion of the fuel pellet remain unchanged

during this investigation, and as a result, they will not be covered here. This investigation is









concerned with the basic theoretical calculation of thermal expansion, which is discussed in the

following section.

2.4.2 Fuel Thermal Expansion Subroutine

The subroutine within FRAPCON-3 that directly calculates the thermal expansion of the

fuel is FTHCON.F, which is a subroutine called by FEXPAN.F in the FRAPCON source code.

S= 10i- xT (3.0 x103) + (4.0 x 10-2)e(6.9x0-20/kbxT)
(1.11)

This relationship describes the non-geometry dependent temperature-induced strain of

UO2, and is part of the FRACAS model. The thermal expansion study gives an indication of the

sensitivity of the thermal expansion coefficient while maintaining the methodology for

simulation of other macroscopic properties, and will continue to be studied as the MD model

continues to be developed.

2.5 FRAPCON Code Description

This section presents an overview of the particular code structure of FRAPCON.

Because FRAPCON is a product of over 50 years of development and modification, it contains

over 200 modularized and independent subroutines. For this reason, it is favorable for the

selection of particular subroutine modification, without causing catastrophic compilation errors,

provided proper variable control is maintained.

FRAPCON consists of 3 major packages: the main FRPCON package, which calls all

other packages and subroutines, the FRACAS-I package, which calls all components of the

FRACAS-I mechanical model, and the MATPRO materials properties package. For the

purposes of this investigation, both the FRACAS-I and MATPRO packages are to be modified,

while the FRPCON package itself remains unchanged, to maintain fluidity. The hierarchy and









loops structure of the major code components is given in Figure 2-6, modified from the

FRAPCON manual.3

The output of FRAPCON-3 is in the form of a substantial text file, giving thermal,

mechanical, and pressure response data for each time/burnup step. The output file is organized

into three sections: summary-page, axial-region printout, and power/time step printout.

Boundary conditions for the beginning of each time step are explicitly printed into each

power/time step section. Because so many calculations are performed at each radial node, axial

node, region, and time step, the output files for each FRAPCON run are around 500 pages of

text. In order to de-convolute this output, a plot package is included, which collects and plots 1-

D and 2-D data, with selectable fields for the abscissa and ordinate. This plotting package is

currently in the form of a MS excel spreadsheet, and contains both original data retrieval macros,

and some modified for this project.

2.6 FRAPCON Input File

The input file used by FRAPCON is a text file explicitly defining a number of physical

parameters, operation descriptors, evaluation model options, special flags, and isotopic

distributions. Like many FORTRAN IV input files, a FRAPCON input file uses special

characters and commands to evaluate the problem, read appropriate parameters, and create a

robust output file.

For this investigation, it is important to note that the primary focus is to obtain a

quantitative relationship between the unmodified FRAPCON program and a new, hybridized

FRAPCON-AM/MD program for the same fuel rod. For this reason, the input file should have

the same physical parameters in both the FRAPCON and FRAPCON-AM/MD cases. This is, in

fact, the case, with only the output data fields being different for each case. The input file is

included in Appendix A.









In order to establish a base case, an input file was chosen which models a typical fuel rod.

The rod used for this investigation is Oconee Rod 15309, from the Oconee nuclear station near

Greenville, South Carolina. Oconee is a three-unit nuclear power station owned by Duke Energy

Corporation Company and operated by Duke Power Company which produced over 19 million

megawatt hours of electricity per year as of 2003.10 Unit 1 began operations in 1973, and the

plant is licensed to operate until 2033. All three units are supplied by Babcock and Wilcox (now

AREVA ANP).

For this investigation, 34 time steps are used, representing a burnup of 0 to 46.2

Megawatt days per kilogram Uranium (MWd/kgU), over a lifetime of 1550 days. Table 2-1

summarizes of the major parameters of the pellet in the FRAPCON input file:

Table 2-1 FRAPCON input parameters for Oconee rod model.
Number of time steps 34
Number of axial regions 12
Number of equi-volume radial rings 15
Pellet Density 95% theoretical density (td=10.96g/cm3)
Enrichment 3% (atom %)
Pitch 1.4224 cm


In addition to the parameters listed on Table 2-1, it should be noted that no gadolinia was

included in the fuel, and the fuel is presumed to be fresh, unirradiated U02. The MOX fuel case

is also available in FRAPCON, but has not yet been investigated against an appropriate AM/MD

model.




























DetennineGap
Conductance


Gas release. void
volumes, pressure


Yes


Figure 2-1. Simplified FRAPCON solution scheme for one time step






















AT = 0 AT = ATf,- ATfs
Figure 2-2. Temperature calculation for 1-D assumption

Crud


Figure 2-3.3 Fuel rod temperature radial distribution










Fuel Centerline




Mesh Points


,1 2 3 4 ...










Figure 2-4. Mesh point layout



6


5-










1
4-

3-

C2 .




0
0 500 1000 1500 2000 2500 3000
Temperature, K

Ronchi Data ------ Lucuta Modified NFI


Figure 2-5.7 FRAPCON conductivity functions with Ronchi data


3500











t t 1 4 T
FRAPCON-3 Local power Calcate
SLocal burnmap Total rod gas
S Gas roduclon Plnm tap
SI as preuIre
I alculate I




Hydrogen uprake

Initaliz C Ilaln inp p I
Inltes Therw.ile.xpan. O
rbles Idiated aroemh Prnt out

calculate --
SFuel pellet relocation
Do i- I NT II
r-- TimeTI I IFRPTR
Sloop Do 1-1, rstr

Somc I pda
Pow er condisos i sculatc el
Coolant coiIios Tenpatres
Caikulate Thenual expansion
Flu-. luenc, u Sclli
coldwork I ina tic Tion


Do a-1,N Claddng (and fuld)
gas Im/e i mechanical response

2 Calculate gapM
: hicas FRAPCON-AMAI/iD
I 2 a Temperature p
Doj-I.J Confldxtatnce Modifications
Sd axial aodeop

Void volumes
C, 3Gas release
Sorfed eier'
,it* ------



Figure 2-6.3 FRAPCON solution schematic











FRPCON ----- SETUP --,-- POINTR

S-- INITIAL ---- PRINT L

--- AXIEF

STORE

I
r 1-, + -. -
r-------------------------

_---BURNTUP
i -- GASPRO
5 i---COOLT
-_ -- FLMDRP
z -- CORROS
C ---CLADRP
-- -- TURBOMTL'BRNP
S r---+------------------
Sw --TMPSUB ----FUELTP
i S -- FEXPAN
Q- L SWTLL
t 45 -- FRACAS
1 I- -NEWGAP
St ---CONDUC
t g n ---+------------------
S-- -VOLUME
t t-- FGASRE -----MASSIH
-- -GASREL
L ._ ._. _. L ._. _.. ._. _. .... .. .
I-- FRACAS
L--PLNT
!-- GSPRES
t L -------- -- .----_ ___ -___ __---------
PRINT2
L-- C CREEP
L_------------- ------------------
-- PRINT


Figure 2-7. FRAPCON calling sequence. (boxed names are modified in AM/MD).









CHAPTER 3
MOLECULAR DYNAMICS MODELLING

The molecular dynamics model used in this investigation (AM/MD) will eventually

describe components not necessarily included in a more generic MD case. Generally, MD

models have been developed to describe statistical fluid mechanics, borrowing from

mathematics, physics, and chemistry. As a way to produce a simulated experiment, of sorts, it

has been called "statistical mechanics by numbers"."The AM/MD model presented in this

investigation will eventually include an array of micro-structure components, both intrinsic and

as a result ofburnup. Phenomena such a fission gas production, fission product migration,

chemistry/stoichiometry changes, and grain growth will be manifested as specific point-defects

represented design parameters created in the AM/MD model. In this way, these fission-related

phenomena, previously accounted for by empirical data correlations, can be represented by first-

principles design in a never before seen manner.

3.1 Molecular Dynamics Simulation

Molecular-dynamics (MD) simulation is a well-developed computational methodology,

widely used in the materials and physics communities for the simulation of the properties and

behavior of materials with regard to stoichiometry and point defect changes.12'13 In essence,

MD simulation allows a finite number of particles to interact based on known laws of physics,

and then uses the results to determine the overall properties of the system. Because of the

complexity of molecular systems, it is often impossible to obtain materials properties by purely

analytical methods. For this reason, MD simulation uses numerical methods to determine

thermo-mechanical properties of a given system based on input parameters including time and

position.









Traditionally, the procedure for MD simulation has been to examine a material at the

lattice level, with the atoms treated as classical objects described by Newton's equations of

motion. Each individual atom moves according to the net force exerted on it from all of the other

atoms in the system; these forces are described in terms of an interaction potential, which is a

relationship for the dependence of the energy and force on an atom due to the other atoms in the

system. The mathematical form of the interatomic potential depends on the material simulated

and is typically parameterized to experimental properties, such as the crystal structure, lattice

parameterss, elastic properties and other pertinent physical properties.

With the advent of more capable computational systems, MD simulations have become

more efficient. Typically, these MD simulations model up to millions of atoms, allowing small,

but finite amounts of material to be simulated. The effects of temperature and the dynamical

evolution of the system can be captured in this approach in a natural way. More particularly,

As described in detail in additional MD publications from the University of Florida Department

of Materials Science and Engineering, 14,15,16 MD simulations have been used to characterize the

thermal expansion and thermal conductivity of pure, defect-free U02. The results from such

simulations are used in this investigation as modification to the FRAPCON source code.

3.2 The AM/MD Model Results

Based on the studies currently being carried out by the University of Florida Materials

Science and Engineering Department, a number of MD simulations have been undertaken. For

this investigation, the MD results of that study yielded specific best fit relationships for both the

thermal conductivity and thermal expansion. These results have yielded a best fit equation for

the instantaneous thermal conductivity given by Equation 2-1.

59848
ko 59848 T1291)(2-1)
(2.38)









Equation 2-1 is used as the basis for calculating the simple thermal conductivity as a

function of temperature. For the nominal temperature ranges associated with steady-state reactor

operations (up to 1400 K), it is known that phonon-phonon interaction is the dominant form of

heat transfer. The physical meaning of the single term in Equation 2-1 represents this phonon

interaction, and it is expressed in Debye form: k=(1/a+bT).17 At higher temperatures, effects of

polarons become increasingly significant, but for the purposes of this investigation, the polaron

interactions are not included in the AM/MD model. As is evident from the equation, no

realtionship between burnup and conductivity is given. This relationship is not yet explored by

the AM/MD. Instead, the AM/MD will simply be substituted for equation 1-3 in the FRAPCON

code. For the investigation of thermal expansion, the AM/MD model predicted

S= 8.835 x10-6T 2.65 x 103 (2-2)

According to the AM/MD simulation, the predicted thermal expansion does not change to

any great extent for systems with point defects or polycrystalline microstructures. This is

consistent with the FRACAS assumption of the independence of the thermal expansion on

microstructure, and with experimental results on a wide range of materials.















Figure 3-1. Typical U02 lattice for MD simulation









CHAPTER 4
SIMULATION METHODOLOGY

In essence, two virtually identical FRAPCON performance codes are created. The first,

which provides a base case, is the original FRAPCON-3 source code, with all original

conductivity and thermal expansion calculation algorithms in place. When this collection of

source code it compiled, the compiler creates an executable FRAPCON file, which is initiated by

the FRAPCON input file. The second case, the FRAPCON-AM/MD performance code is the

project containing the modified thermal conductivity and expansion algorithms. In order to

compare the reactor performance of the original FRAPCON and FRAPCON-AM/MD models,

the same input file is used. This section explains the computational integration of the AM/MD

model with the FRAPCON model

In order to effectively implement the results of the AM/MD simulation into the

FRAPCON-3 code, it is important to establish continuity between the explicit definitions of the

quantities to be modeled. In the case of thermal conductivity, it is important that because the

AM/MD model predicts thermal conductivity only for fresh, un-irradiated fuel, that the

FRAPCON simulation be run with a substitution for base conductivity, but that the embedded

burnup and degradation effects not be changed. For thermal expansion, it is important that a

normalized stress strain start point be established, so that FRAPCON has a known reference

point for making the temperature-dependent stress/strain calculations required for each time step.

The following sections explain how this implementation was carried out.

4.1 Thermal Conductivity FRAPCON/AMMD Implementation

The AM/MD equation established for thermal conductivity (equation 2-1) was compared

to that of the Modified NFI thermal conductivity model given in the latest FRAPCON-3 release

(equation 1-3), and it was found that in the pertinent temperature range of typical LWR









operations, there is relative consistency with both the magnitude and shape of the

temperature/conductivity curve. While it is evident that the AM/MD model predicts a slightly

higher conductivity up to 950K, and then a slightly lower conductivity at higher temperatures,

these results would indicate that the model is sufficiently close for implementation into the

FRAPCON-3 code. Figure 4-1 gives a plot of the conductivity predicted by FRAPCON/NFI

method plotted against that obtained using the AM/MD simulation.

The source code file which controls the instantaneous base conductivity (prior to burnup

phenomena modification) is a subroutine named FTHCON.F, and is called by the main

FRPCON.F package which returns an instantaneous thermal conductivity value to the

FRPCON.F routine. The input variables for this routine include current stage burnup,

temperature of the radial ring, modified fuel density, and stoichiometry ratio (atoms-

metal/atoms-oxide), producing an output of the instantaneous modified thermal conductivity.

The value calculated for this conductivity is then output back to the overall thermal conductivity

iterative loop, where burnup effects discussed in Section I.C.2. are manifested. The FTHCON.F

source file can be found in its entirety in Appendix A. An excerpt of the modified FTHCON.F

subroutine is given in figure 4-2.

4.2 Thermal Expansion FRAPCON-AM/MD Implementation

In order to properly implement the thermal expansion calculations associated with the

AM/MD simulation, it is important to note the definitions given within both the FRAPCON code

and AM/MD values. Within FRAPCON, a number of manifestations of thermal-induced stress

and strain are given, including strains associated with components outside the fuel. For the

cladding strain, no modifications were made to the code in order to maintain overall consistency

with the two simulations. In the fuel pellet itself, there are a number of calculations carried out









by the FRACAS-I model, including axial strain, hoop strain, radial strain. In addition, the surface

displacement and accumulated lifetime strain are also included.

In the case of the AM/MD model, there is no sense of dimensionality or U02 pellet shape;

thus, the concept of thermal-induced strain is given in the context only of a change in the lattice

parameter. While it is possible to assume equiaxial growth in each dimension, such as in the

case of isotropic thermal expansion, it remains to be evaluated just how valid of an assumption

this will ultimately lead to. At this point in the investigation, the lattice parameter expansion

derived from the AMMD model will be substituted directly for the linear growth due to thermal

expansion in FRAPCON.

The temperature-dependent lattice parameter expansion (equation 2-2), with a conversion

into linear strain, as well as a zero-intercept adjustment to 300K derived from the AMMD

simulation is given by equation 3-1 and plotted against that given in the FRACAS-I model

(equation 1-11), and given in figure 4-3.

This plot shows that the two models exhibit similar behavior, but are not in quantitative

agreement. This disagreement notwithstanding, it is useful to carry out a preliminary analysis of

the effect of the thermal expansion model on the evolution of the system during burnup. The

apparent inconsistency for predictions of the thermal expansion are a result of the rather poor

materials fidelity of the interatomic potential used in the AM/MD simulations. While the

AM/MD model is based on the work by Yamada, MD models more consistent with geometrical

effects, as well as refinement of interatomic potentials with respect to stoichiometric changes and

defect formation must be implemented in the future. For the proof of principle presented in this

investigation, however, the AM/MD model will be used to produce a sensitivity analysis

indicative of the effect of fuel thermal expansion on overall fuel performance. The









implementation of the thermal expansion model required slightly more changes be made within

the FRAPCON code. Because the FRACAS-I model uses the thermal expansion to calculate

macroscopic thermo-mechanical phenomena, it was important to modify only the thermal-

induced swelling of the fuel, so that it could be integrated into FRAPCON as seamlessly as

possible, and without interfering with the calculation of other thermo-mechanical phenomena

already in place. To do this, the subroutine FEXPAN.F was modified to calculate thermal

expansion from the AM/MD thermal expansion (equation 2-2). The FEXPAN.F subroutine is

available in Appendix B.

While the thermal expansion properties of UO2 are well known, this sensitivity analysis

study is paramount in this investigation, because it allows for a radically independent model to

describe a phenomenon, and gives the potential to do so for other materials which are as of yet,

not as well known.


20
S18 MATPRO
g 18
S-- AM/MD
S16 -
S14
12 -
i10 -






0
U 6





200 700 1200 1700 2200
Temperature (K)


Figure 4-1. Thermal conductivity predicted by FRAPCON and AM/MD calculations.












*deck fthcon


subroutine fthcon (ftemp,fraden,fotmtl,con,burnup
+ ,gadoln,imox)
c fthcon calculates the fuel thermal conductivity and its
c derivative with respect to temperature as a function of
c temperature, density, composition and burnup.
c uo2 Fuel (IMOX = 0)
c The equation used in this subroutine is that proposed by
c staff at NFI, Japan, at the May 1997 ANS Topical Meeting on
c Light Water Reactor Fuel performance in Portland, OR: (Ohira,
c K., and N.Itagaki, 1997. "Thermal conductivity Measurements
c of High Burnup UO2 Pellet and a Benchmark calculation of Fuel
c Center Temperature", proceedings pp. 541-549. Applies to UO2.
c burnup = current local burnup (Mwd/MTU)
c con = output fuel thermal conductivity (w/(m*K))
c ftemp = current fuel ring temperature (K)
c fraden = input fuel density (ratio of actual density to
c theoretical density)
c fotmtl = input oxygen to metal ratio of fuel (atoms oxygen/
c atoms metal)
c gadoln = input weight fraction of gadolinia in the fuel
c
c the following inputs are by common block
c comp = input puo2 content of fuel (percent puo2
c in total fuel weight)
c bu = input burnup (mw-s/kg-u)
verify subroutine entered
cwrite (*,*) 'Entered fthcon.f'
c
c find constants:
c frpu = comp/100.
t = ftemp
c
c Burnup in GWd/MTU
c
bug = burnup/1000.0
c
if(imox.eq.0) then
c
h = 1/(1.0+396.0*exp(-6380.0/t))
rphonon= 1.0/(0.0452+0.000246*t + 1.0*0.00187*bug+1.1599*gadoln
& + (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h)
elect = (3.50e9/t**2)*exp(-16361/t)

CCORRELATION GIVEN BY AM/MD (3/2/2007)
C
base=59848*ftemp**(-1.291)/2.38

fm = fraden/(1.0 + 0.5*(1.0-fraden))
con = base*fm*1.079

cCORRELATION GIVEN BY FINK ----> NOTE: FOR 95% DENSE
C
c con = 100/(6.548+25.533*(ftemp/1000))
c & + 6400/((ftemp/1000)**(5/2))*exp(16.35/(ftemp/1000))

c
c find uncertainty
if(imox.eq.0) then
if(t.lt.ftmelt) then
ucon = 0.2*(1.0+abs(2.0-fotmtl)*10.)
else
ucon = con/2.0
endif
else
if(t.le.1800.0) then
ucon = 0.07*con
else
frac=(t-1800.0)/(3100.0-1800.0)*(0.20-0.07)+0.07
ucon=frac*con
endif
endif
if (emflag(locidx).eq.on) call emfton (ftemp,fraden,ftmelt,con)
return
end


Figure 4-2. Modified FTHCON.F code excerpt









0.014
.....--- -- FRAPCON
0.012 -- AMMD

0.010 ,*

0 0.008 .,

0.006 ,'

0.004 /

0.002

0.000
0 400 800 1200 1600
Temperature (K)

Figure 4-3 Temperature-dependent strain for FRACAS-I and AM/MD. (relative to 300K).









CHAPTER 5
RESULTS AND OUTLOOK

5.1 FRAPCON-AM/MD Conductivity Results

The first and most direct method by which to analyze the AM/MD model implanted into the

FRAPCON code is to analyze the Beginning of Life (BOL) temperature profile at full power.

This situation has a real world analogy to the situation in which a reactor comes online for the

very first time at full power, with burnup essentially zero.

In order to do this, data was mined from the output file given by the FRAPCON executable,

and plotted using a data extraction macro in Microsoft Excel. Because FRAPCON uses a 1-D,

multi node approach, each axial and radial node distinct temperature distributions. The For the

sake of simplicity, the limiting case was taken. For both the FRAPCON and FRAPCON-

AM/MD cases, the 'hottest' node was axial node 4 of 12. The corresponding output obtained for

the full power/zero-burnup cases, with corresponding heat generation rates, is given in Appendix

D. Figure 5-1 compares the radial temperature profiles for the FRAPCON and FRAPCON-

AM/MD cases. It is clear from this figure that while the centerline temperature is slightly lower

for the AM/MD case (i.e, the conductivity is slightly overestimated), the temperature distribution

is very consistent in both cases. Such a result would be expected based on the fact that the

FRAPCON model contains similar thermal conductivity function obtained by the AM/MD

model.

Perhaps the most encouraging part of this finding is that the interatomic potentials used in the

AM/MD have been successful in calculating the thermal conductivity of fresh UO2 fuel in a

radically independent way from that used in FRAPCON, yet yield nearly identical fuel

performance. Because of the nearly identical chemical properties of uranium and plutonium,

UO2 and PuO2 have the same fluorite crystal structures, similar melting points, etc. For this









reason, the possibility exists to manipulate the AM/MD model in such a way as to either

systematically or randomly substitute Pu cations for U cations, adding another set of interatomic

potentials, and thus creating a MOX fuel simulation with very little additional effort. In the

future, it is expected that other, more exotic fuels, such as SiC and other inert matrix fuels will be

modeled using MD methods.

5.2 FRAPCON-AM/MD Thermal Expansion Results

After restoring the original FRAPCON conductivity and modifying the FEXPAN.F

subroutine, the same procedure as that done with the thermal conductivity was carried out.

Namely, that two executable files were created; one for the FRAPCON case, and one for the

FRAPCON-AM/MD case. For this comparison, it was determined that a good indication of the

sensitivity of the system to thermal-induced expansion could be evaluated in the context of both

instantaneous and burnup-dependent conditions. More particularly, fuel relocation and

irradiation-induced swelling become increasingly significant compared to thermal-induced

swelling as the life cycle of the fuel proceeds. This is evident in the apparent convergence of

surface displacement calculations near 900 Full power days (-27 MWd/kgU).

The results from 5-2 show the densification of fuel for the first few bumup steps, as well

as the prescribed bumup effects evident from empirical correlations given in FRAPCON.

While two models exhibit similar behavior, they are not in quantitative agreement. This

disagreement notwithstanding, this preliminary analysis of the effect of the thermal expansion

model on the evolution of the system during burnup is useful in determining a qualitative feel for

the degree to which bumup phenomena vs. thermal expansion affect the strain conditions of the

fuel pellet. Rigorous studies of the nature of these bumup phenomena have been done, and are

continuing to be developed. It is a major ultimate goal of this AM/MD project to be able to

simulate these burnup effects from a first-principles standpoint using MD models. Such a model









would incorporate not only stoichiometry conditions and interatomic potentials, but

transmutation radiativee capture), grain boundary restructuring, fission gas release, high energy

radiation particle damage, and their relationships to defect formation. Macroscopic phenomena,

such as geometrical concerns and fuel flaking/cracking present unique challenges that may not

be suitable for MD modeling. Such questions remain to be determined by further research on this

AM/MD project.

5.3 Proposed AM/MD Modifications

It is clear that in order to rectify the issues with thermal expansion, a more in-depth

AM/MD model will be required. Fortunately, the integration of such a model has been proven to

be a straightforward process, and it can be taken into consideration piece by piece. It has been

shown in a recent study (Govers, 2007) that current interatomic potentials lack the fidelity

required to compete with detailed experiments 1. A primary step in the development of this

model will be the continued development of higher fidelity potentials, which can be integrated

into the code. A second major step in development of this project is the rigorous evaluation of

proper infiltration ofMD models into FRAPCON. This will come with an increased emphasis on

developing MD models in a functional form capable of coupling prudently with FRAPCON.

The third and ultimate proposals for the AM/MD model is the incorporation of degradation

and burnup phenomena, as well as radiation and fission event effects in the context of MD. A

model must be created such that in intact lattice suffers a displacing fission event, creating high

energy fission products as well as secondary radiation particles, lattice vacancies, interstitials,

stoichiometric changes, production of fission gasses/porosity and fuel swelling/densification. By

assigning a weight determined through sensitivity analysis to each of these phenomena, either a

numerical or a Monte Carlo model be created to simulate the radiative capture and/or fission of a

single atom. Figure 5-3 shows a purely conceptual example of how a fission event may change a









lattice. With such a model created, the avenue will be set for creating MD-based inert matrix fuel

performance calculations.

5.4 Conclusions and Recommendations

In summary, this work is successful in providing provides a proof-of-principle that a fuel

performance code can be adapted to accept input from atomistic model simulations. A very

significant amount of further research is, however, required before atomistic models can reliably

supplement, let alone overtake accuracy obtained by detailed experiments. As mentioned in the

previous section, the direct comparisons with base FRAPCON results show current AM/MD

models cannot yet provide inputs of sufficient materials fidelity to be quantitatively predictive.

Such materials fidelity is only achievable by continued research in improvement of interatomic

potentials.

It is also clear that AM/MD model agrees far better with the FRAPCON's overall

prediction of BOL characteristics than it does with the simple thermal conductivity model for

UO2 itself. For the thermal expansion the rather significant deviation of the AM/MD model from

the FRAPCON model is not manifested in the surface displacement for the first 100 days, after

which, issues related to code implementation may become more profound. These issues will be

addressed in continued development of this project.

The two properties addressed in this investigation are likely the most simple to integrate

into a fuel performance code, but are in addition the most important. The sizable challenge faced

in developing an atomistically informed fuel-performance code is that of incorporating the

complexities associated with the changing chemistry associated with such effects as fuel

swelling, fuel densification, fuel relocation, cladding thermal expansion, cladding creep,

cladding plasticity, and fission gas release.









In more general and far-reaching terms, the development of multi-scale models such as the

integrated FRAPCON-AM/MD is intended to ultimately help optimize the nuclear fuel materials

selection process through atomic level molecular dynamics (MD) involving first-principles

materials simulations. It is a goal of this investigation that a robust method by which nuclear

fuel will be selected can be created by understanding performance sensitivities exposed by MD

simulation. Incorporation of such a process will minimize the need for expensive and time-

consuming in-reactor experimental testing.


1 4CIO

1200

1000

800

600


0 0.1 0.2 0.3 0.4 0.5
Raidus(cm)


Figure 5-1. Pellet temperature profile for MATPRO and AM/MD cases at BOL











20

18

16

14


S10

8
6
i4
as


0 4
0



Figure 5-2.


S,,i 1000
Fuli Power Days


Lifetime surface displacement of fuel pellet


K New Particle Creation






Altered Interatomic Potentials


Figure 5-3. Conceptual model of fission event and lattice effects


1600












APPENDIX A
FTHCON.F SOURCE FILE

*deck fthcon

subroutine fthcon (ftemp,fraden,fotmtl,con,burnup
+ ,gadoln,imox)
C
c fthcon calculates the fuel thermal conductivity and its
c derivative with respect to temperature as a function of
c temperature, density, composition and burnup.
c
c uo2 Fuel (IMOX = 0)
c
c The equation used in this subroutine is that proposed by
c staff at NFI, Japan, at the May 1997 ANS Topical Meeting on
c Light Water Reactor Fuel performance in Portland, OR: (Ohira,
c K., and N.Itagaki, 1997. "Thermal conductivity Measurements
c of High Burnup UO2 Pellet and a Benchmark calculation of Fuel
c Center Temperature", proceedings pp. 541-549. Applies to UO2.
c
c MOX:
c
c Option number 1 (IMOX = 1)
c
c The 100% dense solid MOX fuel thermal conductivity formulation is based
c on a combination of the Duriez stoichiometry-dependent correlation,
c derived from diffusivity measurements on unirradiated fuel pellets
c (C.Duriez, et al, J.Nuclear Materials 277, 143-158 2000) and the burnup
c degradation conatined in a modified version of the NFI fuel thermal
c conductivity model
c
c option number 2 (IMOX = 2)
c
c The MOX fuel thermal conductivity formulation is based
c on the OECD Halden Reactor Project report "Thermal Performance of
c of High Burnup Fuel In-pile Temperature Data and Analysis"
c w.wiesnack, T. Tverberg, Proceedings of the 2000 International
c Topical Meeting on LWR Fuel Performance
c
c
c *********MODIFICATION: Daniel vega (2006/2007)**********************
c
c option number 3 (IMOX = 3)
c
c For now, the MOX fuel thermal conductivity formulation will be given
c an arbitrary and unrealistic value. This will be used to verify that
c FRAPCON will still run properly when IMOX is neither 1 nor 2. i.e., that
c IMOX is of the correct data type to accept 1, 2, or 3.
c
c
C
c burnup = current local burnup (Mwd/MTU)
c con = output fuel thermal conductivity (w/(m*K))
c ftemp = current fuel ring temperature (K)
c fraden = input fuel density (ratio of actual density to
c theoretical density)
c fotmtl = input oxygen to metal ratio of fuel (atoms oxygen/
c atoms metal)
c gadoln = input weight fraction of gadolinia in the fuel
c
c the following inputs are by common block
c comp = input puo2 content of fuel (percent puo2
c in total fuel weight)
c bu = input burnup (mw-s/kg-u)
c emflag(12) = input switch for evaluation model. if this
c variable is equal to 1.0, the matpro model for
c fuel thermal conductivity is replaced by the
c subcode emfton
c

common / phypro / ftmelt,fhefus,ctmelt,chefus,ctranb,
+ ctrane,ctranz,fdelta,bu,comp,deloxy
c
include 'lacmdl.h'
data on / 1 /,
+ off / 2 /,
+ locidx / 12
c












c Verify subroutine entered
c write (*,*) 'Entered fthcon.f'
c
c find constants
c
frpu = comp/100.
t = ftemp
c
c Burnup in GWd/MTU
c
bug = burnup/1000.0
C
if(imox.eq.0) then
C
c NFI formula (Ohira & Itagaki, ANS LWR Fuel perf. Topical mtg. 1997)
c MODIFIED in January 2002 to raise low-burnup thermal conductivity
c at low temperature and lower thermal conductivity at very high temp.
c
h = 1/(1.0+396.0*exp(-6380.0/t))
rphonon= 1.0/(0.0452+0.000246*t + 1.0*0.00187*bug+1.1599*gadoln
& + (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h)
elect = (3.50e9/t**2)*exp(-16361/t)

C STANDARD FRAPCON HALDEN CORRELATION
c base = rphonon + elect
c


C CORRELATION GIVEN BY TAKU WATANABE (3/2/2007)
C
base=59848*ftemp**(-1.291)/2.38
C
C

fm = fraden/(1.0 + 0.5*(1.0-fraden))
con = base*fm*1.079

c CORRELATION GIVEN BY FINK ----> NOTE: FOR 95% DENSE
C
c con = 100/(6.548+25.533*(ftemp/1000))
c &+ 6400/((ftemp/1000)*(5/2))*exp(-16.35/(ftemp/1000))


c fm is the Lucuta porosity correction factor(applied to 100% TD fuel)
c

c
c NFI base equation is for 95% TD fuel, so multiply by 1.079 to
c raise to 100% TD fuel conductivity, then multiply by fm
c

c
else if(imox.eq.1) then
c write (*,*) 'IMOX = 1'
c
c Using the Duriez/NFI Mod correlation combination
c
c base term for MOX
c where x = deviation from stoichiometry (2-O/M)
fm = 1.0789*fraden/(1.0+0.5*(1.0-fraden))
c fm is multiplied by 1.0789 to account for 95% TD
c Porosity correction is Lucuta correction, not Maxwell-Euken
c as proposed by Duriez et al.
x = 2.0-fotmtl
ax=2.85*x+0.035
cx=(2.86-7.15*x)*1.0e-4
c
h = 1/(1.0+396.0*exp(-6380.0/t))
rphonon = 1.0/(ax + cx*t + 0.00187*bug+1.1599*gadoln
&+ (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h)
elect = (1.50e9/t**2)*exp(-13520/t)
base = rphonon + elect
con = base*fm
c
else if(imox.eq.2) then
c
c using the Halden correlation
c
tc=t-273.15
tco=min(1650.0,tc)
buguo2=bug*0.8815












fm = 1.0789*fraden/(1.0+0.5*(1.0-fraden))


base=0.92/(0.1148+0.004*buguo2+1.1599*gadoln+
& 2.475e-4*(1.0-0.00333*buguo2)*tco)+
& 0.0132*exp(0.00188*tc)

con=base*fm
C
c else if (imox.eq.3) call newmodule(ftemp,fraden,fotmtl,con,burnup,gadoln)
else if(imox.eq.3) then
call newmod(ftemp,fraden,fotmtl,con,burnup,gadoln)
c
c NFI formula (Ohira & Itagaki, ANS LWR Fuel perf. Topical mtg. 1997)
c MODIFIED in January 2002 to raise low-burnup thermal conductivity
c at low temperature and lower thermal conductivity at very high temp.
C
h = 1/(1.0+396.0*exp(-6380.0/t))
rphonon= 1.0/(0.0452+0.000246*t + 1.0*0.00187*bug+1.1599*gadoln
& + (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h)
elect = (3.50e9/t**2)*exp(-16361/t)
base = rphonon + elect
c
c fm is the Lucuta porosity correction factor(applied to 100% TD fuel)
c
fm = fraden/(1.0 + 0.5*(1.0-fraden))
c
c NFI base equation is for 95% TD fuel, so multiply by 1.079 to
c raise to 100% TD fuel conductivity, then multiply by fm
c
con = base*fm*1.079
c write(*,*) 'cond = ', con
c arbitrary value insertion for con (W/m-K)
c con=l
c
c If IMOX.ne.0,1,2,3 then stop the calculations
c
else
stop 'fthcon IMOX not within bounds'
end if
c write(*,*) 'cond = ', con
c
c find uncertainty
if(imox.eq.0) then
if(t.lt.ftmelt) then
ucon = 0.2*(1.0+abs(2.0-fotmtl)*10.)
else
ucon = con/2.0
endif
else
if(t.le.1800.0) then
ucon = 0.07*con
else
frac=(t-1800.0)/(3100.0-1800.0)*(0.20-0.07)+0.07
ucon=frac*con
endif
endif
if (emflag(locidx).eq.on) call emfton (ftemp,fraden,ftmelt,con)
return
end












APPENDIX B
FEXPAN.F SOURCE FILE

*deck fexpan
c
subroutine fexpan (ftmelt,sumexp,nr,nrml,tfr,tfring,uo2exp
+ ,afal,crad,j,na,dph)
c implicit real 8 (a-h,o-z)
C
c fexpan is called from frpcon and computes the therm exp of fuel
c this subroutine was coded by g a berna in march 1978.
c
c input arguments
c ********************************************************************
c afal additional thermal expansion factor
c ftmelt fuel melt temperature (K)
c j axial node index
c nr maximum number of radial nodes
c na number of axial nodes plus one
c nrml nr 1
c crad cold state radii of fuel radial nodes (in)
c tfring fuel ring temperatures (F)
c tfr radial node temperatures (F)
c
c output arguments
c ********************************************************************
c dph thermally expanded pellet diameter (in)
c sumexp total fuel surface displacement due to thermal expan.(in)
c uo2exp thermal expansion (in/in)
c
real radn, alphaT
c CURRENTLY THERE ARE 17 RADIAL NODES --> nr = 17
dimension tfr(50) ,tfring(50) ,crad(50) ,uo2exp(50,21)
c write(*,*) 'nr: ', nr
sumexp = O.eO
do 100 l=1,nrml
tfring(l) = 0.5*(tfr(1) + tfr(l+l))
tfringk = (tfring(l) + 459.67)/1.8 !convert F to K
facmot= 0.0
if (tfringk.gt.ftmelt) facmot = 1.0

C *******************ORIGINAL*******************
c write(*,*) tfringk
uo2exp(l,j-1) = fthexp(tfringk,facmot) afal

c
c **********************************************
C **************NERI MOD**************
c write(*,*) afal
c uo2exp(l,j-1) = 8.835e-6*tfringk-2.650497690e-3
c **************NERI MOD**************

c **************FINK'S MOD**************
c if (tfringk.le.923) then
c uo2exp(l,j-1) = (9.828e-6)-(6.93e-10)*tfringk+(1.33e-12)
c &*tfringk**2-(1.757e-17)*tfringk**3
c else if (tfringk.gt.923) then
c uo2exp(l,j-1) = (1.1833e-5)-(5.013e-9)*tfringk+(3.756e-12)
c &*tfringk**3-(6.125e-17)*tfringk**3
c end if
c uo2exp(l,j-1) = uo2exp(l,j-l)*tfringk
c **************FINK'S MOD**************

c write(*,*) 'temp: ', tfringk,'th.exp: ',uo2exp(l,j-1)

c write(*,*) 'tfring=',tfring(l),'FRAP=',uo2exp(l,j-1)
c alphaT=8.835*tfring(l)*0.000001
c write(*,*) 'MD=',alphaT, 'ratio: ', uo2exp(l,j-l)/alphaT

sumexp=sumexp+(crad(l)-crad(l+1))*(1.e0+uo2exp(l,j-1))
100 continue

c write(*,*) 'uo2exp', uo2exp
radn=dph/2
sumexp = sumexp + crad(nr) (1.eO + uo2exp(nrml,j-1))
c write(*,*) 'temp(1): ', tfr(1),'rado: ',crad(1),
c &'radn', radn
dph = sumexp 2.e0
return
end












APPENDIX C
FRAPCON AND FRAPCON-AM/MD INPUT FILE

*****************************************************************************
* frapcon3, steady-state fuel rod analysis code, version 1
A----------------------------------------------------------------------

* CASE DESCRIPTION: Test Case Oconee Rod 15309

*UNIT FILE DESCRIPTION
*---- ---------------------------------------- Output:
* Output :
* 6 STANDARD PRINTER OUTPUT

* Scratch:
* 5 SCRATCH INPUT FILE FROM ECH01

* Input: FRAPCON2 INPUT FILE (UNIT 55)
*****************************************************************************
* GOESINS:
FILE05='nullfile', STATUS='scratch', FORM='FORMATTED',
CARRIAGE CONTROL='LIST'

* GOESOUTS:
FILEO6='test.out', STATUS='UNKNOWN', CARRIAGE CONTROL='LIST' *test
FILE66='test.plot', STATUS='UNKNOWN', CARRIAGE CONTROL='LIST'
/****************************************************************************
Oconee rod 15309
$frpcn
im=34, na=12,
ngasr = 15,
$end
$frpcon
cpl = 10.5, crdt = 0.2, crdtr = 0.0, thkcld = 0.0265,
dco = 0.430, pitch = 0.56,
den = 95., thkgap=0.0050, dishsd = 0.050,dspg = 0.37,
dspgw = 0.055, enrch = 3., fa= 1.0, fgpav = 480,
hplt = 0.70, hdish = 0.014, icm = 4,
icor = 0, idxgas = 1, imox = 0, plant =-2, iq = 0, jdlpr = 0,
total= 11.75, jn = 13,13,13,13,13, jst = 7*1,10*2,2*3,5*4,10*5
rc = 0.0, roughc = 1.97e-5, nplot = 1,
roughf = 2.36e-5, vs = 20.0,
nunits = 1, rsntr = 150.,
qf(1)=0.2,1.0,1.2,1.25,1.25,1.22,1.2,1.16,1.14,1.06,.78,.3, .15,
qf(14)=0.2,1.08,1.18,1.12,1.04,0.97,0.97,1.00,1.03,1.05,1.10,0.97,0.2,
qf(27)=0.2,0.82,1.02,1.11,1.13,1.08,1.04,1.05,1.14,1.19,1.13,0.9,0.2,
qf(40)=0.2,0.95,1.05,1.03,1.03,1.08,1.12,1.12,1.1,1.05,1.0,0.81,0.4,
qf(53)=0.45,0.94,1.02,1.05,1.07,1.10,1.12,1.11,1.10,1.06,1.02,0.95,0.5
x(1)=0,1,2,3,4,5,6,7,8,9,10,11,11.75
x(14)=0,1,2,3,4,5,6,7,8,9,10,11,11.75
x(27)=0,1,2,3,4,5,6,7,8,9,10,11,11.75
x(40)=0,1,2,3,4,5,6,7,8,9,10,11,11.75
x(53)=0,1,2,3,4,5,6,7,8,9,10,11,11.75
flux = 13*0.25e17, p2(1) = 2200.0, tw(1) = 555.0, go(l) = 2.6e6,
ProblemTime= 0.1,65,125,185,210,235,295,
325,350,360,370,500,510,535,540,560,600,
615,850,
890,905, 920,1130,1150,
1160,1205,1220,1240,1400,1445,1490,1510,1535,1550,
qmpy = 5.8,5.8,7.9,7.5,7.3,6.8,6.6,
7.9,7.6,7.4,6.9.6,6,6.1,6.7,6.0,6.6,6.1,
4.1, 5.4,
5.1,4.7,5.4,5.0,4.5,
4.3,4.4,4.3,4.4,4.5,4.55,4.6,4.65,4.7,3.6,
slim = .05,
Send














APPENDIX D

BOL RADIAL TEMPERATURE DISTRIBUTIONS


D.1 FRAPCON BOL Radial Temperature Profile (node 4/12):


avg. linear heat rating
local linear heat rating
6.97E+05(2.21E+05)
peak linear heat rating
step starts at time, day
time increment, day
end step, day


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x **** FRAPCON-3.3 (Aug. 12 05) **** x
x Released August 2005 x
x Oconee rod 15309 x
x run date: 07-oct-30 page 7 x
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

axial region number 4 power-time step 1
kw/m(kw/ft) 19.03( 5.80)


kw/m(kw/ft) 23.94( 7.30)

kw/m(kw/ft) 23.93( 7.29)


s(sec) 0.00( 0
s(sec) 0.10( 8
s(sec) 0.10( 8


fuel-- center
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel-- outer surface

cladding inner sur
cladding outer sur
oxide surface

coolant temperature


rod surface heat flux, W/m**2(btu/h


.OOE+00) starting burnup, MWd/kgU(MWd/mtU)
.64E+03) burnup increment, MWd/kgU(MWd/mtU)
.64E+03) end step burnup, MWd/kgU(MWd/mtU)
Radial temperature and power distribution
Radii cm(in) Temperature, K(F)
0.00000 ( 0.00000) 1348.5 (1967.6
0.08331 ( 0.03280) 1327.1 (1929.2
0.15618 ( 0.06149) 1274.6 (1834.6
0.21931 ( 0.08634) 1205.8 (1710.E
0.27339 ( 0.10763) 1131.9 (1577.E
0.31913 ( 0.12564) 1060.4 (1449.1
0.35724 ( 0.14064) 995.7 (1332.6
0.38841 ( 0.15292) 939.9 (1232.2
0.41336 ( 0.16274) 893.9 (1149.3
0.43278 ( 0.17039) 857.3 (1083.5
0.44738 ( 0.17613) 829.6 (1033.6
0.45783 ( 0.18025) 809.6 ( 997.5
0.46483 ( 0.18300) 796.1 ( 973.3
0.46908 ( 0.18468) 787.9 ( 958.5
0.47126 ( 0.18554) 783.7 ( 951.C
0.47207 ( 0.18585) 782.1 ( 948.2
ce 0.47218 ( 0.18590) 781.9 ( 947.E
face 0.47942 ( 0.18875) 628.0 ( 670.E
face 0.54691 ( 0.21532) 597.9 ( 616.E
0.54692 ( 0.21532) 597.9 ( 616.5


ir-ft**2)


0.00( 0.)
0.00( 4.)
0.00( 4.)


Power profile

6) 0.9682
) 0.9701
6) 0.9750
) 0.9817
) 0.9893
1) 0.9970
6) 1.0043
) 1.0110
) 1.0168
5) 1.0216
6) 1.0254
5) 1.0282
) 1.0301
5) 1.0313
)) 1.0320
2) 1.0324
8) 1.0325
)
)
5)


573.3 ( 572.3)


e

















D.2. FRAPCON-AM/MD BOL Radial Temperature Profile (node 4/12):


x **** FRAPCON-3.3 (Aug. 12 05) **** x
x Released August 2005 x
x Oconee rod 15309 x
x run date: 07-Oct-31 page 8 x
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


axial region number 5

avg. linear heat rating, kw/m(kw/ft) 19.03( 5.80)
local linear heat rating, kw/m(kw/ft) 23.70( 7.22)
6.90E+05(2.19E+05)
peak linear heat rating, kw/m(kw/ft) 23.93( 7.29)


step starts at time, days(sec)
time increment, days(sec)
end step, days(sec)


0.00( 0.OOE+00)
0.10( 8.64E+03)
0.10( 8.64E+03)


power-time step 1


rod surface heat flux, W/m**2(btu/hr-ft**2)


starting burnup, MWd/kgU(MWd/mtU)
burnup increment, MWd/kgU(MWd/mtU)
end step burnup, MWd/kgU(MWd/mtU)


0.00( 0.)
0.00( 4.)
0.00( 4.)


Radial temperature and power distribution


fuel-- center
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel--
fuel-- outer surface

cladding inner surface
cladding outer surface

oxide surface


Radii cm(in)

0.00000 ( 0.00000)
0.08332 ( 0.03280)
0.15621 ( 0.06150)
0.21934 ( 0.08635)
0.27341 ( 0.10764)
0.31915 ( 0.12565)
0.35725 ( 0.14065)
0.38842 ( 0.15292)
0.41337 ( 0.16274)
0.43279 ( 0.17039)
0.44738 ( 0.17613)
0.45783 ( 0.18025)
0.46483 ( 0.18301)
0.46908 ( 0.18468)
0.47126 ( 0.18554)
0.47207 ( 0.18585)
0.47218 ( 0.18590)

0.47943 ( 0.18875)
0.54692 ( 0.21532)

0.54693 ( 0.21533)


Temperature, K(F) Power profile


1364.6 (1996.6)
1339.8 (1951.9)
1279.7 (1843.8)
1203.3 (1706.3)
1123.9 (1563.4)
1049.7 (1429.7)
984.5 (1312.5)
929.9 (1214.2)
885.9 (1135.0)
851.6 (1073.3)
826.0 (1027.0)
807.6 ( 994.1)
795.4 ( 972.1)
788.0 ( 958.8)
784.2 ( 952.0)
782.8 ( 949.4)
782.6 ( 949.1)

630.6 ( 675.5)
600.9 ( 621.9)

600.8 ( 621.8)


0.9682
0.9701
0.9750
0.9817
0.9893
0.9970
1.0043
1.0110
1.0168
1.0216
1.0254
1.0281
1.0301
1.0313
1.0320
1.0324
1.0325


coolant temperature


576.6 ( 578.3)









LIST OF REFERENCES


1Pacific NorthwestNationalLaboratory (1997). FRAPCON-3: A Computer Code for the
Calculation of Steady-State, Thermal-mechanical Behavior of Oxide Fuel Rods. Pacific
Northwest National Laboratory.

2 U.S. Department of Energy (2006) The Global Nuclear Energy Partnership Statement of
Principles. http://www.gnep.energy.gov/

3 Berna, G. A., Beyer, C.E., Davis, K.L., Lanning, D.D. (1997). FRAPCON-3: A computer Code
for the Calculation os Steady-State, Thermal-Mechanical Behavior of Oxide Fuel Rods for High
Burnup. U.S. Nuclear Regulatory Commission. Washington, D.C., NUREG/CR-6534.

4 IdahoNational_Laboratory (2005). Relap5. Relap-5. Idaho Falls, Idaho National Laboratory:
Hydrodynamics and Reactor Kinetics Code.

5 D. D. Lanning, C. E. B., K.J.Geelhood (2005). FRAPCON-3 Updates, Including Mixed-Oxide
Fuel Properties. U. S. N. R. Commission. Washington, D.C., Pacific Northwest National
Laboratory. vol 4.

6 Ohira, K., Itagaki, N. (1997). Thermal Conductivity Measurements of High Burnup UO2 Pellet
and a Benchmark Calculation of Fuel Center Temperature ANS International Topical Meeting on
LWR Fuel Performance, Portland, Oregon, American Nuclear Society.

7 Lucuta, P. G., H.S. Matzke, and I.J. Hastings (1996). "A Pragmatic Approach to Modeling
Thermal Conductivity of Irradiated UO2 Fuel: Review and Recommendations." Journal of
Nuclear Materials 232: 166-180.

8 Ronchi, C., M. Sheindlin, M. Musella, and G.J. Hyland. (1999.). "Thermal Conductivity of
Uranium Dioxide Up to 2900K from Simultaneous Measurement of the Heat Capacity and
Thermal Diffusivity." Journal of Applied Physics 85(2): 776-789.

9 Bohn, M. P. (1977). "FRACAS: A Subcode for the Analysis of Fuel Pellet-Cladding
Mechanical Interaction." TREE-NUREG-1028.

10 NRC. (2004). "U.S. Nuclear Power Plants: Oconee."
http://www.eia.doe.gov/cneaf/nuclear/page/ata_glance/reactors/oconee.html.

11 Schlick, T. (1996). Pursuing Laplace's Vision on Modern Computers. IMA Volumes in
Mathematics and Its Applications New York, Springer-Verlag. 82: 218-247.

12 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press,
1987).

13 D. Frenkel and B. Smit, Molecular Dynamics Simulation, 2nd ed. (Academic Press, 2002).









14 T. Watanabe, S. B. Sinnott, J. S. Tulenko, R. W. Grimes, P. K. Schelling, and S. R. Phillpot, J.
Nucl. Mater., (in print) (2008).
15 T. Watanabe, S. G. Srivilliputhur, P. K. Schelling, J. S. Tulenko, S. B. Sinnott and S. R.
Phillpot, J. Am. Cer. Soc., (in preparation) (2008).

16 Watanabe, T. (2008). Thermal Transport in Uranium Dioxide and Diamond by Atomic Level
Simulations. Department of Materials Science and Engineering. Gainesville, University of
Florida. Ph.D Dissertation: 269.

17N. W. Ashcroft and D. N. Mermin, Solid State Physics (Saunders College Publishing, 1976).

18K. Govers, S. Lemehov, and M. Verwerft, J. Nucl. Mater. 266, 161 (2007).









BIOGRAPHICAL SKETCH

Daniel Vega graduated from Franklin High School in El Paso, TX in 2000, and enrolled at

Texas A&M University. While an undergraduate, Daniel studied physics and nuclear

engineering, worked as a reactor operator at the Texas A&M Nuclear Science Center, and spent

a year working on international nuclear safeguards for the International Atomic Energy Agency

(IAEA) in Vienna, Austria. He graduated with a B.S. in nuclear engineering in December 2005,

and pursued his maritime interests as a marine assistant on a National Science Foundation (NSF)

funded marine geophysics cruise in Antarctica. Then, Mr. Vega enrolled as a graduate student at

the University of Florida in the Department of Nuclear and Radiological Engineering, and has

recently received an M.S. in nuclear engineering. He has accepted a position with the U.S.

Department of Energy's Office of Nuclear Energy, and begins work in August 2008. His

interests include outdoor activities, travel, music, and his dog, Sandwich.





PAGE 1

1 ATOMISTICALLY INFORMED FUEL PE RFORMANCE CODES: A PROOF OF PRINCIPLE USING MOLECULAR DYNAM ICS AND FRAPCON SIMULATION By DANIEL ARMANDO VEGA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008

PAGE 2

2 2008 Daniel Armando Vega

PAGE 3

3 To Sandwich, my patient cohort

PAGE 4

4 ACKNOWLEDGMENTS I would like to thank Professo r Tulenko, Dr. Phillpot, Dr. Si nnott, Taku Wantanabe, and the res t of the research team th at contributed the work, guidance, criticism, and pa tience required to make this research possible. In addition, I extend my appreciat ion to the U.S. Department of Energys Nuclear Engineering Research Ini tiative (NERI) award se lection committee for acknowledging the merit of this i nvestigation and awarding necessary research funds. Lastly, I would like to thank the University of Florid as Nuclear and Radiol ogical Engineering and Materials Science Departments for this joint venture.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4LIST OF TABLES ...........................................................................................................................7LIST OF FIGURES .........................................................................................................................8LIST OF ABBREVIAT IONS AND TERMS .................................................................................. 9ABSTRACT ...................................................................................................................... .............10 CHAP TER 1 INTRODUCTION .................................................................................................................. 112 FRAPCON-3 FUEL PERF ORM ANCE CODE .....................................................................132.1FRAPCON Objectives ................................................................................................. 132.2FRAPCON-3 Solution Description..............................................................................132.3Fuel Rod Thermal Conductivity Response Calculation ...............................................142.3.1Fuel Pellet Thermal Energy Di stribution and Conduction Model .................... 152.3.2FRAPCON Thermal Conductivity Function ....................................................172.4 Fuel Rod Thermal Expansion and Mechanical Response ............................................ 202.4.1FRACAS-I Deformation Model .......................................................................212.4.2Fuel Thermal Expansion Subroutine ................................................................ 222.5FRAPCON Code Description ...................................................................................... 222.6FRAPCON Input File .................................................................................................. 233 MOLECULAR DYNAMICS MODELLING ........................................................................303.1Molecular Dynamics Simulation ................................................................................. 303.2The AM/MD Model Results ........................................................................................314 SIMULATION METHODOLOGY ....................................................................................... 334.1Thermal Conductivity FRAPCON/AMMD Implementation ...................................... 334.2Thermal Expansion FRAPCON-AM/MD Implementation .........................................345 RESULTS AND OUTLOOK ................................................................................................. 395.1FRAPCON-AM/MD Conductivity Results .................................................................395.2FRAPCON-AM/MD Thermal Expansion Results .......................................................405.3Proposed AM/MD Modifications ................................................................................415.4Conclusions and Recommendations ............................................................................42

PAGE 6

6 APPENDIX A FTHCON.F SOURCE FILE ................................................................................................... 45B FEXPAN.F SOURCE FILE ................................................................................................... 48C FRAPCON and FRAPCONAM/MD Input File ....................................................................49D BOL RADIAL TEMPERATURE DISTRIBUTIONS ........................................................... 50D.1FRAPCON BOL Radial Temperat ure Profile (node 4/12): .........................................50D.2.FRAPCON-AM/MD BOL Radial Temp erature Profile (node 4/12): .........................51LIST OF REFERENCES ...............................................................................................................52BIOGRAPHICAL SKETCH .........................................................................................................54

PAGE 7

7 LIST OF TABLES Table page 2-1 The FRAPCON input parameters for the Oconee rod m odel. ........................................... 24

PAGE 8

8 LIST OF FIGURES Figure page 2-1 Simplified FRAPCON solution scheme for one time step ................................................ 25 2-2 Temperature calcula tion for 1-D assumption .................................................................... 262-3 Fuel rod temperatur e radial distribution ............................................................................262-4 Mesh point layout ......................................................................................................... .....272-5 FRAPCON conductivity functi ons with Ronchi data ........................................................272-6 FRAPCON solution schematic .......................................................................................... 282-7 FRAPCON calling sequen ce. (boxed names are modified in AM/MD). .......................... 293-1 Typical UO2 lattice for MD simulation ............................................................................. 324-1 Thermal conductivity predicted by FRAPCON and AM/MD calculations. ......................364-2 Modified FTHCON.F code excerpt ...................................................................................374-3 Temperature-dependent strain for FRAC AS-I and AM/MD. (relative to 300K). ............ 385-1 Pellet temperature profile for MATPRO and AM/MD cases at BOL ...............................435-2 Lifetime surface displacement of fuel pellet ...................................................................... 445-3 Conceptual model of fissi on event and lattice effects ....................................................... 44

PAGE 9

9 LIST OF ABBREVIATIONS AND TERMS BOL Beginning of life: freshly fabricated fuel, zero burnup. Burnup A pseudo-intensive property descri bing the amount thermal energy released per unit mass of nuclear fuel. Typically given in units of megawatt days per kilogram of Uranium (MWd/kg U) CRUD Chalk river unidentif ied deposits: corrosion a nd wear products (rust particles, etc.) that become radioact ive (i.e., activated) when exposed to radiation. EOL End of life: time/burnup level associ ated with the end of a fuel rods lifetime, at which point it is removed from the core and becomes spent fuel LWR Light water reactor: a thermal nuclear reactor that uses ordinary water (light water) as both its neutron moderato r and coolant. T ypically, either Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). MD Molecular dynamics Phonon A quantized lattice vibration, modeled as a means for heat transfer in a rigid lattice Polaron A quasi-particle composed of an electron plus its accompanying polarization field. At high temperatures, becomes significant in heat transfer PW R Pressurized Water Reactor: a nuclear reactor that uses a pressure vessel around the core, keeping it under high pressure to prevent water in the primary cooling loop from boiling. Mo st reactor worldwide are PWRs. IMF Inert matrix fuel

PAGE 10

10 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Master of Science ATOMISTICALLY INFORMED FUEL PE RFORMANCE CODES: A PROOF OF PRINCIPLE USING MOLECULAR DYNAM ICS AND FRAPCON SIMULATION By Daniel Armando Vega August 2008 Chair: James S. Tulenko Major: Nuclear Engineering Sciences Current limitations on Uranium di oxide fuel lifetime are determined largely from physical degradation of the fuel pellets. In order to better understand th e behavior of Uranium dioxide fuel, new atomic level approaches are being used to simulate th e effects of both temperature and burnup. A proof-of-principle study is presented in which the results of atomic-level simulations of the thermal expansion and ther mal conductivity of Uranium dioxide are integrated into the Light Water Reactor fuel performance code FRAPCON. The beginning of life (BOL) thermal conductivity profile of a fu el pellet, and the evolution of the pellet expansion over its lifetime are calculated. It is found that (i) modifying FRAP CON in such a way as to accept input from the results obtained through atomistic simulations by the integration of at omistic models into FRAPCON is possible for a number of thermo-mech anical properties, and (ii) the properties determined from atomistic simulations yiel d predictions in FRAP CON that are in good agreement for the BOL thermal conductivity, but less satisfactory for the pellet thermal expansion, due to phenomena thus far unaccounted for by atomic level simulations. Still, by successfully incorporating these simulations in to FRAPCON, a basis for a more complete atomistic model is built, demonstrating the potential for more sophisticated and encompassing first principles models.

PAGE 11

11 CHAPTER 1 INTRODUCTION Having been in operation for over 35 years, LW Rs using uranium dioxide (UO2) fuel have been modeled and studied using a variety of neutronic, thermo-hydraulic, and thermomechanical approaches. Traditionally, the deve lopment of such simulations has been guided by studying the macroscopic phenomen a characterized in laboratory experiments and during commercial reactor operations. A key part of the development and qualification of UO2 fuel systems consists of analysis in fuel performance codes such as FRAPCON.1 FRAPCON-3, the most recent release, encapsu lates the behavior of UO2 fuel pellets in terms of specific thermomechanical phenomena and carefully determined and validated empirical relationships. This approach has been extraordinarily successful in providing quantification of fuel properties. However, the development of the key empirical relationships is extremely time-consuming and expensive. In addition, such empirical relationships are often valid only for di screte cases, creating simulations with proprietary modules; thus, a ny approaches that could reduce the time and expense involved while extending the scope of validity of the models would be extremely valuable. In this investigation, the first steps in using results from atomistic models, specifically molecular-dynamics simulations, directly as mo dels into FRAPCON-3, is explored. The first step in this investigation is to study the natu re and structure of FRAPCON-3 to determine the most prudent sections of s ource code to modify for atomistic simulation. Secondly, an appropriate description of th e Atomistic/Molecular Dynamics Model (AM/MD) is presented. Then, a discussion of the coupled simulation me thodology is presented, in which the integration of the FRAPCON and AM/MD is explained. Fi nally, the results obtai ned by this coupled simulation are presented for analysis and recommendations.

PAGE 12

12 It is important to note that while atomic-lev el simulation can be powerful, the modeling of every component of a nuclear core is substantial a nd outside the scope of th is investigation; thus atomic level simulation for this project pertains only to the UO2 fuel pellets, keeping the thermal transport of the gap, cladding, and coolant unc hanged. In addition, the fuel rod-specific macroscopic parameters by which fuel performance is based, is composed of a complex set of relationships describing geometry-dependent phenomena, material region interfaces, and composition changes. At this point in the AM/MD model, many of these phenomena have not yet been described. However, the most fundame ntal aspects of thermo-mechanical properties modeling, namely the thermal conductivity and ther mal expansion, are appr opriate candidates for rigorous first-principles modeling. By first establishing and valid ating these radically different models by comparing them to established models, th e avenue for first princi ples investigations of yet to be modeled fuel lattices becomes widene d. Such a bridge from empirical to firstprinciples models is a theme central to this investigation. This research is funded by DOE-NERI Awa rd DE-FC07-05ID14649, as part of the overall Global Nuclear Energy Partnership (GNEP) initiative.2

PAGE 13

13 CHAPTER 2 FRAPCON-3 FUEL PERFORMANCE CODE 2.1 FRAPCON Objectives Funda mentally, the FRAPCON series was create d to accurately calculate the performance of LWR fuel rods3: a major objective of the reactor safety research program sponsored by the U.S. Nuclear Regulatory Commission (NRC). As a result of an ex tensive analytical performance code development program, FRAPCON was created as a joint effort between Idaho National Engineering and Environmental Laboratory (IN EEL) and Pacific Northw est National Laboratory (PNNL). INEEL has since become Idaho National Laboratory (INL), and both facilities are now operated by Battelle. With the creation of such a code, the NRC has the ability to not only achieve the object of accurately calculating thermo-mechanical pr operties and performance of fuel rods, but to independently investigate de sign parameters put fo rth by both nuclear vendors and utilities without us ing proprietary performance codes. By using both in-reactor and out-ofreactor experiments to benchmark and asses the code capabilities, FRAPCON has proven to be a reliable tool for simple analysis of LWR fuel rod performance. It is important to note that FRAPCON is a steady state code which refers to the situati on where (i) short-term changes are sufficiently small that they can be considered constant for a discreet time step, and (ii) boundary conditions for each iteration do not change. This analysis is valid for constant power and slow up-ramp or down-ramp operations typical of normal commercial reactor operations. Reactor accidents and transient an alysis are included in FRAPTRAN, and not discussed in this investigation. 2.2 FRAPCON-3 Solution Description In order to evaluate steady state fuel rod performance, FRAPCON-3 uses an iterative process based on defined boundary c onditions at each tim e step. Simply put, the interrelated

PAGE 14

14 effects of fission rate, thermal energy distributio n, fuel and cladding temp erature, fission product release, fuel swelling/densif ication, irradiation-induced growth, thermal expansion, crud deposition, cladding corrosion are coupled t ogether and run through calculation loops until temperature solutions converge to within an acce ptable error range. Na turally, some of these phenomena are fuel rod power/temperature specific while others are cumulative, and heavily dependent on burnup and previous conditions. For th is reason, a collection of loops is used to calculate the properties for each time step. Becau se the number and length of time steps is user defined, it is apparent that increased accuracy can be achieved with finer time steps, while computational demand can be assuaged by more coar se time steps. Much of the information in this section is adapted from the FRAPCON manual,3 and the components most pertinent to this investigation are included in this section. At the beginning of each time step, a number of operations are carried out: 1. Initial fuel rod data is determined by input from previous step (or initial condition) 2. Temperature of fuel a nd cladding are computed 3. Fission product generation & release, internal gas pressure computed 4. Temperature response is iterated through loop unt il solution converges to less than 1% of T When the temperature converges, the parameters are all record ed, sent to the output file, and used for the initial conditions for the next time step. Figure 2-1 gives a simplified solution scheme flow chart. 2.3 Fuel Rod Thermal Conductivity Response Calculation To further simplify heat transport calculati ons, each calcu lation for the fuel pellet temperature distribution is taken in such a way that heat transfer is only in the radial direction. This is a valid assumption if one assumes (i) azimuthal homogeneity (axissymmetric analysis),

PAGE 15

15 and (ii) for each axial node, heat transfer in the axial direction is negligible as compared to that in the radial direction, and is thus ignored (Figure 2-2). The centerline temperature of th e pellet, being the innermost and hottest component, is the first temperature calculated, followed by the pe llet surface temperatur e, the inner cladding temperature, the outer cladding temperature, th e oxidation layer temperat ure, and finally, the bulk coolant temperature. In simple terms, the te mperature change from the fuel centerline to the bulk coolant can be given in terms of com ponent temperature changes by the equation: f cb u l kf i l mc r u do xc l a dg a pf u e l fc bulk filmT(z)=T(z)+ T(z)+ T(z)+ T(z)+ T(z)+ T(z)+ T(z) where T = fuel centerline temperature T= bulk coolant temperature T= temperature change through the forced convection ficrud ox clad gap fuellm layer T= temperature change through crud layer T= temperature change through oxide Layer T= temperature change through cladding T= temperature change through gap T= temperature change through fuel pellet 2.3.1 Fuel Pellet Thermal Energy Di stribution and Conduction Model While cond uction through each region is importa nt, for the purposes of this section, only the conduction through the pellet is analyzed. This is because the pellet is the only region to which fundamental source code changes are ma de for the FRAPCON-AM/MD investigation. The finite difference (FD) heat conducti on models used on FRAPCON-3 are based on those presented in RELAP-5.4 Instead of the method of weighted residuals used in previous releases of FRAPCON, the FD approach us ed in FRAPCON-3 must contain fine-mesh capabilities for high burnup analysis and must interface with other burnup models either in existence or yet to be created.3 (1-1)

PAGE 16

16 While an FD approach is straightforward for localized or uniform heat sources, the nonuniformity of internal energy production requires that the FD method ta ke into account the spatial dependence of internal heat sources. In addition, because FRAP CON-3 contains a userdefined number of temperature nodes, as well as variable mesh spaci ng, the FD method must allow for such flexibility. The steady-state inte gral form of the heat conduction equation is: (,)()()SVkTxTxnsSxV Where k = thermal conductivity (W/m-K) s = surface of control volume (m2) n = surface normal unit vector S = internal heat source (W/m3) T = temperature (K) V = control volume (m3) x = 1-D space coordinate It is apparent that Equation 1-2 is co mplex enough in 1-D that thousands of FD calculations require a fair amount of computati on. 2-D and 3-D models of this variety would require far greater computation resources. Figure 2-4 shows the 1-D mesh point layout. In order to maintain proper control surface and volume inte grals, it is an important assumption that the geometry remains fixed for this stage in the an alysis. In addition, boundary conditions must be established. For all FD calculations in the fuel pellet, the follow ing boundary conditions are used: 1) 00 xT x 2) Pellet Surface Temperature (Tfs) is defined (1-2) (pellet center is local maximum, azimuthal symetry)

PAGE 17

17 Once a suitable temperature distribution is established, FRAPCON-3 can exit the fuel pellet temperature subroutine and continue with gap, cladding, oxi de, crud, and bulk temperature calculations. When a distribution has been established for the en tire rod, the thermal expansion and stress/strain relationships expressed in othe r subroutines can be eval uated within the same coarse time step. In words, equation 1-2 states that for a steady state fuel pellet, the integrated surface heat flux through the control area into the gap (left hand side) is e qual to the integrated heat generation rate in the control volume (right hand side). With this in mind, the numerical solutions to this equation are based on corre ct balance of temperature and conductivity, equivalent to a heat generation rate. When in corporating an AM/MD mo del, there is no reason to re-evaluate the FD methodology. In fact, in order to discover how the materials properties affect the overall perfor mance of the fuel rod, it is important to maintain code structure for the fuel pellet temperature distribu tion. Returning to equation 1-2, the only parameter that should change is the th ermal conductivity (,) kTxfunction. In order to und erstand where this term comes from, as well as how it compares to an AM/MD model, the following section presents a description of the ther mal conductivity function. 2.3.2 FRAPCON Thermal Co nductivity Function In general, and in the context of FRAPCO N, therm al conductivity (k) has a dependence on temperature. The thermal conductivity function chosen as a base case for use throughout this investigation is and currently used in FRAPCON-3.3 is one pres ented by staff at Nuclear Fuel Industries, Ltd. (Japan), at the May1997 American Nuclear So ciety Topical meeting on Light Water Reactor Fuel Performance.5 The NFI function is of the sa me form as that originally included in FRAPCON3 by Lucata et al.6 Equations 1-3 and 1-4 represent these functions,

PAGE 18

18 respectively. While the functions are similar, the NFI function was chosen, because it is included in the most recent FRAPCON-3 releas e (FRAPCON-3.3). This function has been successfully benchmarked, and prov ides a better fit to collecte d data from a study by Ronchi.8 The study provides a suitable base for t ypical PWR fuel conductivity calculations: -16361 9 () T o 21.0 3.5010 k= ()e (0.0452+0.000246T) T -16361 9 () T o 21.0 4.71510 k= ()e (0.0375+0.0002165T) T where ko = conductivity of unirra diated urania (UO2) T = temperature (K) The gadolinia content for this fuel pellet is 0. The conductivity of unirradiated 95% dense (theoretical) urania thus becomes the starting point for burnup calculations. Figure 2-57 gives a plot of the NFI function, Lucuta function, and experimental data, collected by Ronchi, et al, 1999.8 The Ronchi (1999) data come from a study done on unirradiated pellet material at nominal and high temperatures. A final function describing the conduc tivity used for calculation throughout the time step is given by multiplying ko by a collection of four coefficients representing particular burnup and non-bur nup phenomena. These coefficients are: FD dissolved fission product (t emperature and burnup dependent) FP precipitated fission product (temperature and burnup dependent) FM Maxwell porosity factor ( porosity fraction and shape factor) FR radiation effect (temperature dependent) NFI (1-3) Lucata (1-4)

PAGE 19

19 FD becomes an increasingly significant coeffi cient as temperature and burnup increase, as is evident in its definition: 3.265 3.2651.090.0643 1 arctan 1.090.0643 FD T B B T B B where B = burnup (atom %) 1 atom% = 9.383 GWd/MTU at 200 MeV/fission T = temperature (K) FP also increases with burnup and temperature, and is given by: 1200 1000.0191 1 30.019 1TB FP B e FM, which is related only to porosity, is given by: 1 1(1) p FM s p where p = porosity fraction (as fabricated plus swelling) s = shape factor (1.5 for spheres) FR, which is temperature-dependent and always applied, is given by: 900 800.2 1 1TFR e (1-5) (1-6) (1-7) (1-8)

PAGE 20

20 Finally, these coefficients are put together and multiplied by ko to give a temperatureburnup dependent thermal conductivity value (Equation 1-9) and the FD interval of interest is used for the corresponding temperature, to create a straightforward calculation for instantaneous thermal conductivity of the fuel pellet: ()okkFDFPFMFR This calculation is then carried out for each radial node, after which other FD calculations are performed for the gap, cladding, oxide, and fi lm layers. The thermal distributions for nonfuel regions will not be discussed in this investigation. 2.4 Fuel Rod Thermal Expansion and Mechanical Response Therm al expansion is important to study because its effects lead to structural stress and corresponding phenomena, as well as changing the thermal conductivity coefficient. In addition, proper modeling of thermal expansion is necessary to accurately model other geometrydependent phenomena, such as fission gas release. FRAPCON-3 uses the FRACAS-I9 mechanical model to calculate the fuel and cladding mechanical deformation throughout reactor operation. It is important to note that there are many components to the overall concept of mechanical response. In fact, FRAPCON uses FRACAS-I coupled with other non-specific thermal expansi on models to develop an encompassing solution that describes the state of the fu el rod taking into account geometri cal considerations. It involves a complex set of relationships describing geom etry-dependent phenomena, region interfaces, and composition changes. At this point in the AM/MD model, such macroscopic phenomena are impossible to describe. For this reason, a ll FRAPCON AM/MD calculations will retain the geometry-dependent and burnup correction factor s of FRAPCON, while the nature of the theoretical thermal expansion regimes is chan ged from the empirical FRAPCON model to the (1-9)

PAGE 21

21 first-principles AM/MD model. From a progr amming standpoint, this change may seem minor, but the substitution occurs in the most fundament al and physically signifi cant section of code, and so therefore has the potential to have a profound effect on the overall performance of the code. To illustrate this point, Figure 2-63 gives a detailed flow ch art of the overall FRAPCON process, highlighting the specific sections that are to be modified for the FRAPCON AM/MD model. 2.4.1 FRACAS-I Deformation Model As m entioned previously, the FRACAS-I model is designed to analyze fuel and cladding mechanical deformation in order to better simu late the effects of burnup and temperature on the fuel pin. FRACAS can model two scenarios. The first of these is an open gap situation, when the fuel and cladding are not in co ntact. This open gap situation has cladding with internal and external pressures that must be calculated. The second scenario is when the fuel has expanded enough to be in contact with the cladding. In this closed gap situation, the fuel drives the cladding outwards. Overall, the calculations take into account the effects of fuel thermal expansion, fuel swelling, fuel densificati on, fuel relocation, cladding thermal expansion, cladding creep, cladding plastici ty, and fission gas/external coolant pressures. Because the gap closure is a function of burnup, the FRACAS-I model determines gap thickness as a function of time, and determines the appropria te open-gap or closed-gap model. Fuel expansion leading to a closed gap depends on both thermal factors and radiation factors; thus to properly calculate lifetime strain both the open gap and closed gap models were used for the analysis. At the current stage of analysis, no changes have been made to the gap conductance or fuel cladding regimes in the FRACAS-I model. The com ponents of the FRACAS-I model which are not explicitly used to calculate the theoretical ther mal expansion of the fuel pellet remain unchanged during this investigation, and as a result, they will not be covere d here. This investigation is

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22 concerned with the basic theoretic al calculation of thermal expansion, which is discussed in the following section. 2.4.2 Fuel Thermal Expansion Subroutine The subroutine within F RAPCON-3 that directly calculates the thermal expansion of the fuel is FTHCON.F, which is a subroutine call ed by FEXPAN.F in the FRAPCON source code. -20(6.910/) -5 -3 -210 (3.010) (4.010)bkTTe (1.11) This relationship describes the non-geometry dependent temperature-induced strain of UO2, and is part of the FRACAS model. The ther mal expansion study gives an indication of the sensitivity of the thermal expansion coefficient while maintaining the methodology for simulation of other macroscopic properties, and will continue to be studied as the MD model continues to be developed. 2.5 FRAPCON Code Description This section presents an overview of the particular code structure of FRAPCON. Because FRAPCON is a product of over 50 years of development and modification, it contains over 200 modularized and independe nt subroutines. For this r eason, it is favorable for the selection of particular subrou tine modification, without causing catastrophic compilation errors, provided proper variable control is maintained. FRAPCON consists of 3 major packages: the main FRPCON package, which calls all other packages and subroutines, the FRACAS-I package, which calls all components of the FRACAS-I mechanical model, and the MATPRO materials properties package. For the purposes of this investigation, both the FRACAS -I and MATPRO packages are to be modified, while the FRPCON package itself remains unchange d, to maintain fluidity. The hierarchy and

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23 loops structure of the major code components is given in Figure 2-6, modified from the FRAPCON manual.3 The output of FRAPCON-3 is in the form of a substantial text file, giving thermal, mechanical, and pressure response data for each time/burnup step. The output file is organized into three sections: summary-page, axial-regi on printout, and power/time step printout. Boundary conditions for the beginning of each time step are explicitly printed into each power/time step section. Because so many calculations are performed at each radial node, axial node, region, and time step, the output files for each FRAPCON run are around 500 pages of text. In order to de-convolute this output, a plot package is included, which collects and plots 1D and 2-D data, with selectable fields for the ab scissa and ordinate. This plotting package is currently in the form of a MS ex cel spreadsheet, and contains both original data retrieval macros, and some modified for this project. 2.6 FRAPCON Input File The input file used by FRAPCON is a text file explicitly defining a number of physical parameters, operation descriptors, evaluation model options, special flags, and isotopic distributions. Like many FO RTRAN IV input files, a FRAP CON input file uses special characters and commands to evaluate the problem, read appropriate parameters, and create a robust output file. For this investigation, it is important to note that the primary focus is to obtain a quantitative relationship between the unmodified FRAPCON pr ogram and a new, hybridized FRAPCON-AM/MD program for the sa me fuel rod. For this reason, the input file should have the same physical parameters in both the FRAP CON and FRAPCON-AM/MD cases. This is, in fact, the case, with only the output data fields be ing different for each case. The input file is included in Appendix A.

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24 In order to establish a base case, an input f ile was chosen which models a typical fuel rod. The rod used for this investigation is Ocon ee Rod 15309, from the Oconee nuclear station near Greenville, South Carolina. Oc onee is a three-unit nuc lear power station owned by Duke Energy Corporation Company and operate d by Duke Power Company which produced over 19 million megawatt hours of electric ity per year as of 2003.10 Unit 1 began operations in 1973, and the plant is licensed to operate until 2033. All th ree units are supplied by Babcock and Wilcox (now AREVA ANP). For this investigation, 34 time steps ar e used, representing a burnup of 0 to 46.2 Megawatt days per kilogram Uranium (MWd/kgU ), over a lifetime of 1550 days. Table 2-1 summarizes of the major parameters of the pellet in the FRAPCON input file: Table 2-1 FRAPCON input parameters for Oconee rod model. Number of time steps 34 Number of axial regions 12 Number of equi-volume radial rings 15 Pellet Density 95% theore tical density (td=10.96g/cm3) Enrichment 3% (atom %) Pitch 1.4224 cm In addition to the parameters listed on Tabl e 2-1, it should be noted that no gadolinia was included in the fuel, and the fuel is presumed to be fresh, unirradiated UO2. The MOX fuel case is also available in FRAPCON, but has not yet been investigated agains t an appropriate AM/MD model.

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25 Figure 2-1. Simplified FRAPCON so lution scheme for one time step

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26 Figure 2-2. Temperature calcu lation for 1-D assumption Figure 2-3.3 Fuel rod temperature radial distribution T = 0 T = TfcTfs

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27 Figure 2-4. Mesh point layout Figure 2-5.7 FRAPCON conductivity func tions with Ronchi data 3 1 4 2 Mesh Points Fuel Centerline

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28 Figure 2-6.3 FRAPCON solution schematic

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29 Figure 2-7. FRAPCON calling sequence. (boxed names are modified in AM/MD).

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30 CHAPTER 3 MOLECULAR DYNAMICS MODELLING The m olecular dynamics model used in this investigation (AM/ MD) will eventually describe components not necessarily included in a more generic MD case. Generally, MD models have been developed to describe statistical fluid mechanics, borrowing from mathematics, physics, and chemistry. As a way to produce a simulated experiment, of sorts, it has been called "statistical mechanics by numbers".11The AM/MD model presented in this investigation will eventually incl ude an array of micro-structure components, both intrinsic and as a result of burnup. Phenomena such a fission gas production, fi ssion product migration, chemistry/stoichiometry changes, and grain growth will be manifested as specific point-defects represented design parameters created in the AM/M D model. In this way, these fission-related phenomena, previously accounted for by empirical data correlations, can be represented by firstprinciples design in a never before seen manner. 3.1 Molecular Dynamics Simulation Molecular-dynamics (MD) simulation is a well-developed co mputational methodology, widely used in the materials and physics communities for the simulation of the properties and behavior of materials with regard to stoichiometry and point defect changes.12,13 In essence, MD simulation allows a finite number of particle s to interact based on known laws of physics, and then uses the results to determine the ove rall properties of the system. Because of the complexity of molecular systems, it is often impossible to obtain mate rials properties by purely analytical methods. For this reason, MD simu lation uses numerical methods to determine thermo-mechanical properties of a given system based on input parameters including time and position.

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31 Traditionally, the procedure for MD simulati on has been to examine a material at the lattice level, with the atoms treated as classical objects described by Newtons equations of motion. Each individual atom moves according to th e net force exerted on it from all of the other atoms in the system; these forces are described in terms of an interaction potential, which is a relationship for the dependence of the energy and fo rce on an atom due to the other atoms in the system. The mathematical form of the interato mic potential depends on the material simulated and is typically parameterized to experimental pr operties, such as the crystal structure, lattice parameter(s), elastic properties and other pertinent physic al properties. With the advent of more capable computati onal systems, MD simulations have become more efficient. Typically, these MD simulations model up to millions of atoms, allowing small, but finite amounts of material to be simulated. The effects of temperature and the dynamical evolution of the system can be cap tured in this approach in a natu ral way. More particularly, As described in detail in additi onal MD publications from the Univ ersity of Florida Department of Materials Science and Engineering,14,15,16 MD simulations have been used to characterize the thermal expansion and thermal c onductivity of pure, defect-free UO2. The results from such simulations are used in this investigation as modification to the FRAPCON source code. 3.2 The AM/MD Model Results Based on the studies currently being carried out by the University of Florida Materials Science and Engineering Department, a number of MD simulations have been undertaken. For this investigation, the MD results of that study yielded specific be st fit relationships for both the thermal conductivity and thermal expansion. These results have yielded a best fit equation for the instantaneous thermal conduc tivity given by Equation 2-1. (1.291) o59848 k= (2.38) T (2-1)

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32 Equation 2-1 is used as the basis for cal culating the simple thermal conductivity as a function of temperature. For the nominal temperat ure ranges associated w ith steady-state reactor operations (up to 1400 K), it is known that pho non-phonon interaction is the dominant form of heat transfer. The physical meaning of the si ngle term in Equation 2-1 represents this phonon interaction, and it is expressed in Debye form: k =(1/a+bT).17 At higher temperatures, effects of polarons become increasingly signi ficant, but for the purposes of this investigation, the polaron interactions are not included in the AM/MD model. As is evident from the equation, no realtionship between burnup and conduc tivity is given. This relati onship is not yet explored by the AM/MD. Instead, the AM/MD will simply be substituted for equation 1-3 in the FRAPCON code. For the investigation of therma l expansion, the AM/MD model predicted 3 61065.210835.8 T (2-2) According to the AM/MD simulation, the predicted thermal expansion does not change to any great extent for systems w ith point defects or polycrystal line microstructures. This is consistent with the FRACAS assumption of the independence of the thermal expansion on microstructure, and with experimental re sults on a wide range of materials. Figure 3-1. Typical UO2 lattice for MD simulation

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33 CHAPTER 4 SIMULATION METHODOLOGY In essence, two virtually identical FR APCON performance codes are created. The first, which provides a base case, is the original FRAPCON-3 source code, with all original conductivity and thermal expansion calculation al gorithms in place. When this collection of source code it compiled, the compiler creates an executable FRAPCON file, which is initiated by the FRAPCON input file. The second case, the FRAPCON-AM/MD performance code is the project containing the modified thermal conductiv ity and expansion algorithms. In order to compare the reactor performance of the or iginal FRAPCON and FRAPCON-AM/MD models, the same input file is used. This section explains the computational integration of the AM/MD model with the FRAPCON model In order to effectively implement the results of the AM/MD simulation into the FRAPCON-3 code, it is important to establish continuity between the explicit definitions of the quantities to be modeled. In th e case of thermal conductivity, it is important that because the AM/MD model predicts thermal conductivity onl y for fresh, un-irradiated fuel, that the FRAPCON simulation be run with a substitution for base con ductivity, but that the embedded burnup and degradation effects not be changed. For thermal expansion, it is important that a normalized stress strain start point be esta blished, so that FRAPCON has a known reference point for making the temperature-dependent stress/strain calculations required for each time step. The following sections explain how th is implementation was carried out. 4.1 Thermal Conductivity FRAP CON/AMMD Implementation The AM/MD equation established for therm al conductivity (equation 2-1) was compared to that of the Modified NFI thermal conductivity model given in the la test FRAPCON-3 release (equation 1-3), and it was found that in the pertinent temperature range of typical LWR

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34 operations, there is relative consistency w ith both the magnitude and shape of the temperature/conductivity curve. While it is evident that the AM /MD model predicts a slightly higher conductivity up to 950K, and then a sligh tly lower conductivity at higher temperatures, these results would indicate that the model is sufficiently close for implementation into the FRAPCON-3 code. Figure 4-1 gi ves a plot of the conductivity predicted by FRAPCON/NFI method plotted against that obtai ned using the AM/MD simulation. The source code file which controls the inst antaneous base conductivity (prior to burnup phenomena modification) is a subroutine na med FTHCON.F, and is called by the main FRPCON.F package which returns an instan taneous thermal conductivity value to the FRPCON.F routine. The input variables fo r this routine includ e current stage burnup, temperature of the radial ri ng, modified fuel density, and stoichiometry ratio (atomsmetal/atoms-oxide), producing an output of the instantaneous modified thermal conductivity. The value calculated for this conductivity is th en output back to the overall thermal conductivity iterative loop, where burnup effect s discussed in Section I.C.2. are manifested. The FTHCON.F source file can be found in its entirety in Appe ndix A. An excerpt of the modified FTHCON.F subroutine is given in figure 4-2. 4.2 Thermal Expansion FRAPCON-AM/MD Implementation In order to properly implement the thermal e xpansion calculations associated with the AM/MD simulation, it is important to note the de finitions given within both the FRAPCON code and AM/MD values. Within FRAPCON, a number of manifestations of thermal-induced stress and strain are given, including st rains associated with components outside the fuel. For the cladding strain, no modifications were made to the code in order to maintain overall consistency with the two simulations. In the fuel pellet itse lf, there are a number of calculations carried out

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35 by the FRACAS-I model, including axial strain, hoop strain, radial strain. In addition, the surface displacement and accumulated lifetim e strain are also included. In the case of the AM/MD model, there is no sense of dimensionality or UO2 pellet shape; thus, the concept of thermal-induced strain is give n in the context only of a change in the lattice parameter. While it is possible to assume equi axial growth in each dimension, such as in the case of isotropic thermal expansion, it remains to be evaluated just how valid of an assumption this will ultimately lead to. At this point in the investigation, the la ttice parameter expansion derived from the AMMD model will be substituted directly for the linear growth due to thermal expansion in FRAPCON. The temperature-dependent lattice parameter expansion (equation 2-2), with a conversion into linear strain, as well as a zero-inter cept adjustment to 300K derived from the AMMD simulation is given by equation 3-1 and plotte d against that given in the FRACAS-I model (equation 1-11), and given in figure 4-3. This plot shows that the two models exhibit similar beha vior, but are not in quantitative agreement. This disagreement notwithstanding, it is useful to carry out a preliminary analysis of the effect of the thermal expansion model on the evolution of the sy stem during burnup. The apparent inconsistency for predictions of the th ermal expansion are a result of the rather poor materials fidelity of the interatomic potential used in the AM/MD simulations. While the AM/MD model is based on the work by Yamada, MD models more consistent with geometrical effects, as well as refinement of interatomic poten tials with respect to stoichiometric changes and defect formation must be implemented in the futu re. For the proof of principle presented in this investigation, however, the AM/MD model will be used to produce a sensitivity analysis indicative of the effect of fuel thermal e xpansion on overall fuel performance. The

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36 implementation of the thermal expansion model re quired slightly more changes be made within the FRAPCON code. Because the FRACAS-I mode l uses the thermal expansion to calculate macroscopic thermo-mechanical phenomena, it was important to modify only the thermalinduced swelling of the fuel, so that it could be integrated into FRAPCON as seamlessly as possible, and without interferi ng with the calculation of othe r thermo-mechanical phenomena already in place. To do this, the subroutine FEXPAN.F was modified to calculate thermal expansion from the AM/MD thermal expansion (equation 2-2). The FEXPAN.F subroutine is available in Appendix B. While the thermal expansion properties of UO2 are well known, this sensitivity analysis study is paramount in this investigation, because it allows for a radically independent model to describe a phenomenon, and gives the potential to do so for other materials which are as of yet, not as well known. 0 2 4 6 8 10 12 14 16 18 20 200700120017002200 Temperature (K)Thermal Conductivity (W/m-K) MATPRO AM/MD Figure 4-1. Thermal conductivity predicte d by FRAPCON and AM/MD calculations.

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37 *deck fthcon subroutine fthcon (ftemp,fraden,fotmtl,con,burnup + ,gadoln,imox) c fthcon calculates the fuel thermal conductivity and its c derivative with respect to temperature as a function of c temperature, density, composition and burnup. c UO2 Fuel (IMOX = 0) c The equation used in this subroutine is that proposed by c staff at NFI, Japan, at the May 1997 ANS Topical Meeting on c Light Water Reactor Fuel performance in Portland, OR: (Ohira, c K., and N.Itagaki, 1997. "Thermal Conductivity Measurements c of High Burnup UO2 Pellet and a Benchmark Calculation of Fuel c Center Temperature", proceedings pp. 541-549. Applies to UO2. c burnup = current local burnup (MWd/MTU) c con = output fuel thermal conductivity (W/(m*K)) c ftemp = current fuel ring temperature (K) c fraden = input fuel density (ratio of actual density to c theoretical density) c fotmtl = input oxygen to metal ratio of fuel (atoms oxygen/ c atoms metal) c gadoln = input weight fraction of gadolinia in the fuel c c the following inputs are by common block c comp = input puo2 content of fuel (percent puo2 c in total fuel weight) c bu = input burnup (mw-s/kg-u) c Verify subroutine entered c write (*,*) 'Entered fthcon.f' c c find constants: c frpu = comp/100. t = ftemp c c Burnup in GWd/MTU c bug = burnup/1000.0 c if(imox.eq.0) then c h = 1/(1.0+396.0*exp(-6380.0/t)) rphonon= 1.0/(0.0452+0.000246*t + 1.0*0.00187*bug+1.1599*gadoln & + (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h) elect = (3.50e9/t**2)*exp(-16361/t) ******************************************************* C CORRELATION GIVEN BY AM/MD (3/2/2007) C base=59848*ftemp**(-1.291)/2.38 ******************************************************* fm = fraden/(1.0 + 0.5*(1.0-fraden)) con = base*fm*1.079 ******************************************************* c CORRELATION GIVEN BY FINK ----> NOTE: FOR 95% DENSE c c con = 100/(6.548+25.533*(ftemp/1000)) c & + 6400/((ftemp/1000)**(5/2))*exp(-16.35/(ftemp/1000)) ******************************************************* c c find uncertainty if(imox.eq.0) then if(t.lt.ftmelt) then ucon = 0.2*(1.0+abs(2.0-fotmtl)*10.) else ucon = con/2.0 endif else if(t.le.1800.0) then ucon = 0.07*con else frac=(t-1800.0)/(3100.0-1800.0)*(0.20-0.07)+0.07 ucon=frac*con endif endif if (emflag(locidx).eq.on) call emfton (ftemp,fraden,ftmelt,con) return end Figure 4-2. Modified FTHCON.F code excerpt

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38 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 040080012001600 Temperature (K)Strain (cm/cm ) FRAPCON AM/MD Figure 4-3 Temperature-dependent strain fo r FRACAS-I and AM/MD. (relative to 300K).

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39 CHAPTER 5 RESULTS AND OUTLOOK 5.1 FRAPCON-AM/MD Co nductivity Results The first and m ost direct me thod by which to analyze the AM/MD model implanted into the FRAPCON code is to analyze th e Beginning of Life (BOL) temper ature profile at full power. This situation has a real world analogy to the situation in which a reactor comes online for the very first time at full power, with burnup essentially zero. In order to do this, data was mined from th e output file given by the FRAPCON executable, and plotted using a data extraction macro in Mi crosoft Excel. Because FRAPCON uses a 1-D, multi node approach, each axial and radial node di stinct temperature distributions. The For the sake of simplicity, the limiting case was ta ken. For both the FR APCON and FRAPCONAM/MD cases, the hottest node was axial node 4 of 12. The correspon ding output obtained for the full power/zero-burnup cases, wi th corresponding heat generation rates, is given in Appendix D. Figure 5-1 compares the radial temper ature profiles for the FRAPCON and FRAPCONAM/MD cases. It is clear from this figure that while the centerline temperature is slightly lower for the AM/MD case (i.e, the conductivity is sligh tly overestimated), the temperature distribution is very consistent in both cases. Such a result would be expected based on the fact that the FRAPCON model contains similar thermal conductivity function obtained by the AM/MD model. Perhaps the most encouraging part of this finding is that the interatomic potentials used in the AM/MD have been successful in calcul ating the thermal conductivity of fresh UO2 fuel in a radically independent way from that used in FRAPCON, yet yield ne arly identical fuel performance. Because of the nearly identical chemical properties of uranium and plutonium, UO2 and PuO2 have the same fluorite crystal structures similar melting points, etc. For this

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40 reason, the possibility exists to manipulate the AM/MD model in such a way as to either systematically or randomly substitute Pu cations for U cations, adding another set of interatomic potentials, and thus creating a M OX fuel simulation with very little additional effort. In the future, it is expected that other, more exotic fuels, such as SiC and other inert matrix fuels will be modeled using MD methods. 5.2 FRAPCON-AM/MD Thermal Expansion Results After restoring the original FRAPCON conductivity and modifying the FEXPAN.F subroutine, the same procedure as that done with the thermal conduc tivity was carried out. Namely, that two executable files were created ; one for the FRAPCON case, and one for the FRAPCON-AM/MD case. For this comparison, it was determined that a good indication of the sensitivity of the system to thermal-induced expa nsion could be evaluated in the context of both instantaneous and burnup-dependent conditions. More particularly, fuel relocation and irradiation-induced swelling become increasing ly significant compared to thermal-induced swelling as the life cycle of the fuel proceeds. This is evident in the apparent convergence of surface displacement calculations near 900 Full power days (~27 MWd/kgU). The results from 5-2 show the densification of fuel for the first few burnup steps, as well as the prescribed burnup effects evident from em pirical correlations given in FRAPCON. While two models exhibit similar behavior, they are not in quantitative agreement. This disagreement notwithstanding, this preliminary anal ysis of the effect of the thermal expansion model on the evolution of the syst em during burnup is useful in determining a qualitative feel for the degree to which burnup phenomena vs. thermal e xpansion affect the strain conditions of the fuel pellet. Rigorous studies of the nature of these burnup phenomena have been done, and are continuing to be developed. It is a major ultim ate goal of this AM/MD project to be able to simulate these burnup effects from a first-principles standpoint using MD models. Such a model

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41 would incorporate not only stoichiometry conditions and interatomic potentials, but transmutation (radiative capture), grain boundary restructuring, fi ssion gas release, high energy radiation particle damage, and th eir relationships to defect form ation. Macroscopic phenomena, such as geometrical concerns and fuel flaking/ cracking present unique ch allenges that may not be suitable for MD modeling. Such questions remain to be determined by further research on this AM/MD project. 5.3 Proposed AM/MD Modifications It is clear that in order to rectify the issues with ther m al expansion, a more in-depth AM/MD model will be required. Fortunately, the integration of su ch a model has been proven to be a straightforward process, and it can be taken into consideration piece by piece. It has been shown in a recent study (Govers, 2007) that curr ent interatomic potentia ls lack the fidelity required to compete with detailed experiments18. A primary step in the development of this model will be the continued development of higher fidelity potentials, which can be integrated into the code. A second major step in developmen t of this project is th e rigorous evaluation of proper infiltration of MD models into FRAPCON. This will come with an increased emphasis on developing MD models in a functional form cap able of coupling prudently with FRAPCON. The third and ultimate proposals for the AM/MD model is the incorporation of degradation and burnup phenomena, as well as radiation and fission event effects in th e context of MD. A model must be created such that in intact latt ice suffers a displacing fi ssion event, creating high energy fission products as well as secondary radiation particles, lattice vacancies, interstitials, stoichiometric changes, production of fission gasses/porosity and fuel swelling/densification. By assigning a weight determined th rough sensitivity analysis to each of these phenomena, either a numerical or a Monte Carlo model be created to s imulate the radiative capture and/or fission of a single atom. Figure 5-3 shows a purely conceptual example of how a fission event may change a

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42 lattice. With such a model created, the avenue w ill be set for creating MD-based inert matrix fuel performance calculations. 5.4 Conclusions and Recommendations In summ ary, this work is successful in provi ding provides a proof-of-pr inciple that a fuel performance code can be adapte d to accept input from atomistic model simulations. A very significant amount of further research is, however, required before atomistic models can reliably supplement, let alone overtake accuracy obtained by detailed experiments. As mentioned in the previous section, the direct comparisons with base FRAPCON results show current AM/MD models cannot yet provide inputs of sufficient mate rials fidelity to be quantitatively predictive. Such materials fidelity is only achievable by continued research in improvement of interatomic potentials. It is also clear that AM/MD model agrees far better with th e FRAPCONs overall prediction of BOL characteristics than it does with the simple thermal conductivity model for UO2 itself. For the thermal expansion the rather significant deviation of the AM/MD model from the FRAPCON model is not manifested in the su rface displacement for the first 100 days, after which, issues related to code implementation ma y become more profound. These issues will be addressed in continued deve lopment of this project. The two properties addressed in this investigation are likely the most simple to integrate into a fuel performance code, but are in additio n the most important. The sizable challenge faced in developing an atomistically informed fuel-p erformance code is that of incorporating the complexities associated with the changing chemis try associated with such effects as fuel swelling, fuel densification, fuel relocation, cladding ther mal expansion, cladding creep, cladding plasticity, and fission gas release.

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43 In more general and far-reaching terms, the development of multi-scale models such as the integrated FRAPCON-AM/MD is intended to ultima tely help optimize the nuclear fuel materials selection process through atomic level molecu lar dynamics (MD) involving first-principles materials simulations. It is a goal of this investigation that a robust method by which nuclear fuel will be selected can be created by unders tanding performance sensitivities exposed by MD simulation. Incorporation of such a process will minimize th e need for expensive and timeconsuming in-reactor experimental testing. Figure 5-1. Pellet temperature profile for MATPRO and AM/MD cases at BOL

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44 Figure 5-2. Lifetime surface displacement of fuel pellet Figure 5-3. Conceptual model of fission event and lattice effects

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45 APPENDIX A FTHCON.F SOURCE FILE *deck fthcon subroutine fthcon (ftemp,fraden,fotmtl,con,burnup + ,gadoln,imox) c c fthcon calculates the fuel thermal conductivity and its c derivative with respect to temperature as a function of c temperature, density, composition and burnup. c c UO2 Fuel (IMOX = 0) c c The equation used in this subroutine is that proposed by c staff at NFI, Japan, at the May 1997 ANS Topical Meeting on c Light Water Reactor Fuel performance in Portland, OR: (Ohira, c K., and N.Itagaki, 1997. "Thermal Conductivity Measurements c of High Burnup UO2 Pellet and a Benchmark Calculation of Fuel c Center Temperature", proceedings pp. 541-549. Applies to UO2. c c MOX: c c Option number 1 (IMOX = 1) c c The 100% dense solid MOX fuel thermal conductivity formulation is based c on a combination of the Duriez stoichiometry-dependent correlation, c derived from diffusivity measurements on unirradiated fuel pellets c (C.Duriez, et al, J.Nuclear Materials 277, 143-158 2000) and the burnup c degradation conatined in a modified version of the NFI fuel thermal c conductivity model c c Option number 2 (IMOX = 2) c c The MOX fuel thermal conductivity formulation is based c on the OECD Halden Reactor Project report "Thermal Performance of c of High Burnup Fuel In-pile Temperature Data and Analysis" c W.Wiesnack, T. Tverberg, Proceedings of the 2000 International c Topical Meeting on LWR Fuel Performance c c c *********MODIFICATION: Daniel Vega (2006/2007)********************** c c Option number 3 (IMOX = 3) c c For now, the MOX fuel thermal conductivity formulation will be given c an arbitrary and unrealistic value. This will be used to verify that c FRAPCON will still run properly when IMOX is nither 1 nor 2. i.e., that c IMOX is of the correct data type to accept 1, 2, or 3. c c ******************************************************************** c c burnup = current local burnup (MWd/MTU) c con = output fuel thermal conductivity (W/(m*K)) c ftemp = current fuel ring temperature (K) c fraden = input fuel density (ratio of actual density to c theoretical density) c fotmtl = input oxygen to metal ratio of fuel (atoms oxygen/ c atoms metal) c gadoln = input weight fraction of gadolinia in the fuel c c the following inputs are by common block c comp = input puo2 content of fuel (percent puo2 c in total fuel weight) c bu = input burnup (mw-s/kg-u) c emflag(12) = input switch for evaluation model. if this c variable is equal to 1.0, the matpro model for c fuel thermal conductivity is replaced by the c subcode emfton c common / phypro / ftmelt,fhefus,ctmelt,chefus,ctranb, + ctrane,ctranz,fdelta,bu,comp,deloxy c include 'lacmdl.h' data on / 1 /, + off / 2 /, + locidx / 12 / c

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46 c Verify subroutine entered c write (*,*) 'Entered fthcon.f' c c find constants c frpu = comp/100. t = ftemp c c Burnup in GWd/MTU c bug = burnup/1000.0 c if(imox.eq.0) then c c NFI formula (Ohira & Itagaki, ANS LWR Fuel perf. Topical mtg. 1997) c MODIFIED in January 2002 to raise low-burnup thermal conductivity c at low temperature and lower thermal conductivity at very high temp. c h = 1/(1.0+396.0*exp(-6380.0/t)) rphonon= 1.0/(0.0452+0.000246*t + 1.0*0.00187*bug+1.1599*gadoln & + (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h) elect = (3.50e9/t**2)*exp(-16361/t) ******************************************************* C STANDARD FRAPCON HALDEN CORRELATION c base = rphonon + elect c ******************************************************* ******************************************************* C CORRELATION GIVEN BY TAKU WATANABE (3/2/2007) C base=59848*ftemp**(-1.291)/2.38 C C ******************************************************* fm = fraden/(1.0 + 0.5*(1.0-fraden)) con = base*fm*1.079 ******************************************************* c CORRELATION GIVEN BY FINK ----> NOTE: FOR 95% DENSE c c con = 100/(6.548+25.533*(ftemp/1000)) c & + 6400/((ftemp/1000)**(5/2))*exp(-16.35/(ftemp/1000)) ******************************************************* c fm is the Lucuta porosity correction factor(applied to 100% TD fuel) c c c NFI base equation is for 95% TD fuel, so multiply by 1.079 to c raise to 100% TD fuel conductivity, then multiply by fm c c else if(imox.eq.1) then c write (*,*) 'IMOX = 1' c c Using the Duriez/NFI Mod correlation combination c c base term for MOX c where X = deviation from stoichiometry (2-O/M) fm = 1.0789*fraden/(1.0+0.5*(1.0-fraden)) c fm is multiplied by 1.0789 to account for 95% TD c Porosity correction is Lucuta correction, not Maxwell-Euken c as proposed by Duriez et al. x = 2.0-fotmtl ax=2.85*x+0.035 cx=(2.86-7.15*x)*1.0e-4 c h = 1/(1.0+396.0*exp(-6380.0/t)) rphonon = 1.0/(ax + cx*t + 0.00187*bug+1.1599*gadoln &+ (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h) elect = (1.50e9/t**2)*exp(-13520/t) base = rphonon + elect con = base*fm c else if(imox.eq.2) then c c Using the Halden correlation c tc=t-273.15 tco=min(1650.0,tc) buguo2=bug*0.8815

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47 fm = 1.0789*fraden/(1.0+0.5*(1.0-fraden)) base=0.92/(0.1148+0.004*buguo2+1.1599*gadoln+ & 2.475e-4*(1.0-0.00333*buguo2)*tco)+ & 0.0132*exp(0.00188*tc) con=base*fm c c else if (imox.eq.3) call newmodule(ftemp,fraden,fotmtl,con,burnup,gadoln) else if(imox.eq.3) then call newmod(ftemp,fraden,fotmtl,con,burnup,gadoln) c c NFI formula (Ohira & Itagaki, ANS LWR Fuel perf. Topical mtg. 1997) c MODIFIED in January 2002 to raise low-burnup thermal conductivity c at low temperature and lower thermal conductivity at very high temp. c h = 1/(1.0+396.0*exp(-6380.0/t)) rphonon= 1.0/(0.0452+0.000246*t + 1.0*0.00187*bug+1.1599*gadoln & + (1.0-0.9*exp(-0.04*bug))*0.038*bug**0.28*h) elect = (3.50e9/t**2)*exp(-16361/t) base = rphonon + elect c c fm is the Lucuta porosity correction factor(applied to 100% TD fuel) c fm = fraden/(1.0 + 0.5*(1.0-fraden)) c c NFI base equation is for 95% TD fuel, so multiply by 1.079 to c raise to 100% TD fuel conductivity, then multiply by fm c con = base*fm*1.079 c write(*,*) 'cond = ', con c arbitrary value insertion for con (W/m-K) c con=1 c c If IMOX.ne.0,1,2,3 then stop the calculations c else stop 'fthcon IMOX not within bounds' end if c write(*,*) 'cond = ', con c c find uncertainty if(imox.eq.0) then if(t.lt.ftmelt) then ucon = 0.2*(1.0+abs(2.0-fotmtl)*10.) else ucon = con/2.0 endif else if(t.le.1800.0) then ucon = 0.07*con else frac=(t-1800.0)/(3100.0-1800.0)*(0.20-0.07)+0.07 ucon=frac*con endif endif if (emflag(locidx).eq.on) call emfton (ftemp,fraden,ftmelt,con) return end

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48 APPENDIX B FEXPAN.F SOURCE FILE *deck fexpan c subroutine fexpan (ftmelt,sumexp,nr,nrm1,tfr,tfring,uo2exp + ,afal,crad,j,na,dph) c implicit real 8 (a-h,o-z) c ******************************************************************** c fexpan is called from frpcon and computes the therm exp of fuel c this subroutine was coded by g a berna in march 1978. c ******************************************************************** c input arguments c ******************************************************************** c afal additional thermal expansion factor c ftmelt fuel melt temperature (K) c j axial node index c nr maximum number of radial nodes c na number of axial nodes plus one c nrm1 nr 1 c crad cold state radii of fuel radial nodes (in) c tfring fuel ring temperatures (F) c tfr radial node temperatures (F) c ******************************************************************** c output arguments c ******************************************************************** c dph thermally expanded pellet diameter (in) c sumexp total fuel surface displacement due to thermal expan.(in) c uo2exp thermal expansion (in/in) c ******************************************************************** real radn, alphaT c CURRENTLY THERE ARE 17 RADIAL NODES --> nr = 17 dimension tfr(50) ,tfring(50) ,crad(50) ,uo2exp(50,21) c write(*,*) 'nr: ', nr sumexp = 0.e0 do 100 l=1,nrm1 tfring(l) = 0.5*(tfr(l) + tfr(l+1)) tfringk = (tfring(l) + 459.67)/1.8 !convert F to K facmot = 0.0 if (tfringk.gt.ftmelt) facmot = 1.0 c *******************ORIGINAL******************* c write(*,*) tfringk uo2exp(l,j-1) = fthexp(tfringk,facmot) afal c ********************************************** c **************NERI MOD************** c write(*,*) afal c uo2exp(l,j-1) = 8.835e-6*tfringk-2.650497690e-3 c **************NERI MOD************** c **************FINK'S MOD************** c if (tfringk.le.923) then c uo2exp(l,j-1) = (9.828e-6)-(6.93e-10)*tfringk+(1.33e-12) c &*tfringk**2-(1.757e-17)*tfringk**3 c else if (tfringk.gt.923) then c uo2exp(l,j-1) = (1.1833e-5)-(5.013e-9)*tfringk+(3.756e-12) c &*tfringk**3-(6.125e-17)*tfringk**3 c end if c uo2exp(l,j-1) = uo2exp(l,j-1)*tfringk c **************FINK'S MOD************** c write(*,*) 'temp: ', tfringk,'th.exp: ',uo2exp(l,j-1) c write(*,*) 'tfring=',tfring(l),'FRAP=',uo2exp(l,j-1) c alphaT=8.835*tfring(l)*0.000001 c write(*,*) 'MD=',alphaT, 'ratio: ', uo2exp(l,j-1)/alphaT sumexp=sumexp+(crad(l)-crad(l+1))*(1.e0+uo2exp(l,j-1)) 100 continue c write(*,*) 'uo2exp', uo2exp radn=dph/2 sumexp = sumexp + crad(nr) (1.e0 + uo2exp(nrm1,j-1)) c write(*,*) 'temp(1): ', tfr(1),'rado: ',crad(1), c &'radn', radn dph = sumexp 2.e0 return end

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49 APPENDIX C FRAPCON AND FRAPCON-AM/MD INPUT FILE ***************************************************************************** frapcon3, steady-state fuel rod analysis code, version 1 *---------------------------------------------------------------------* CASE DESCRIPTION: Test Case Oconee Rod 15309 *UNIT FILE DESCRIPTION *--------------------------------------------------Output: Output : 6 STANDARD PRINTER OUTPUT * Scratch: 5 SCRATCH INPUT FILE FROM ECH01 * Input: FRAPCON2 INPUT FILE (UNIT 55) ***************************************************************************** GOESINS: FILE05='nullfile', STATUS='scratch', FORM='FORMATTED', CARRIAGE CONTROL='LIST' GOESOUTS: FILE06='test.out', STATUS='UNKNOWN', CARRIAGE CONTROL='LIST' *test FILE66='test.plot', STATUS='UNKNOWN', CARRIAGE CONTROL='LIST' /**************************************************************************** Oconee rod 15309 $frpcn im=34, na=12, ngasr = 15, $end $frpcon cpl = 10.5, crdt = 0.2, crdtr = 0.0, thkcld = 0.0265, dco = 0.430, pitch = 0.56, den = 95., thkgap=0.0050, dishsd = 0.050,dspg = 0.37, dspgw = 0.055, enrch = 3., fa= 1.0, fgpav = 480, hplt = 0.70, hdish = 0.014, icm = 4, icor = 0, idxgas = 1, imox = 0, iplant =-2, iq = 0, jdlpr = 0, totl = 11.75, jn = 13,13,13,13,13, jst = 7*1,10*2,2*3,5*4,10*5 rc = 0.0, roughc = 1.97e-5, nplot = 1, roughf = 2.36e-5, vs = 20.0, nunits = 1, rsntr = 150., qf(1)=0.2,1.0,1.2,1.25,1.25,1.22,1.2,1.16,1.14,1.06,.78,.3,.15, qf(14)=0.2,1.08,1.18,1.12,1.04,0.97,0.97,1.00,1.03,1.05,1.10,0.97,0.2, qf(27)=0.2,0.82,1.02,1.11,1.13,1.08,1.04,1.05,1.14,1.19,1.13,0.9,0.2, qf(40)=0.2,0.95,1.05,1.03,1.03,1.08,1.12,1.12,1.1,1.05,1.0,0.81,0.4, qf(53)=0.45,0.94,1.02,1.05,1.07,1.10,1.12,1.11,1.10,1.06,1.02,0.95,0.5 x(1)=0,1,2,3,4,5,6,7,8,9,10,11,11.75 x(14)=0,1,2,3,4,5,6,7,8,9,10,11,11.75 x(27)=0,1,2,3,4,5,6,7,8,9,10,11,11.75 x(40)=0,1,2,3,4,5,6,7,8,9,10,11,11.75 x(53)=0,1,2,3,4,5,6,7,8,9,10,11,11.75 flux = 13*0.25e17, p2(1) = 2200.0, tw(1) = 555.0, go(1) = 2.6e6, ProblemTime= 0.1,65,125,185,210,235,295, 325,350,360,370,500,510,535,540,560,600, 615,850, 890,905, 920,1130,1150, 1160,1205,1220,1240,1400,1445,1490,1510,1535,1550, qmpy = 5.8,5.8,7.9,7.5,7.3,6.8,6.6, 7.9,7.6,7.4,6.9,6.6,6.1,6.7,6.0,6.6,6.1, 4.1, 5.4, 5.1,4.7,5.4,5.0,4.5, 4.3,4.4,4.3,4.4,4.5,4.55,4.6,4.65,4.7,3.6, slim = .05, $end

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50 APPENDIX D BOL RADIAL TEMPERATURE DISTRIBUTIONS D.1 FRAPCON BOL Radial Temp erature Profile (node 4/12): xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x **** FRAPCON-3.3 (Aug. 12 05) **** x x Released August 2005 x x Oconee rod 15309 x x run date: 07-Oct-30 page 7 x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx axial region number 4 power-time step 1 avg. linear heat rating, kW/m(kW/ft) 19.03( 5.80) local linear heat rating, kW/m(kW/ft) 23.94( 7.30) rod surface heat flux, W/m**2(btu/hr-ft**2) 6.97E+05(2.21E+05) peak linear heat rating, kW/m(kW/ft) 23.93( 7.29) step starts at time, days(sec) 0.00( 0.00E+00) starting burnup, MWd/kgU(MWd/mtU) 0.00( 0.) time increment, days(sec) 0.10( 8.64E+03) burnup increment, MWd/kgU(MWd/mtU) 0.00( 4.) end step, days(sec) 0.10( 8.64E+03) end step burnup, MWd/kgU(MWd/mtU) 0.00( 4.) Radial temperature and power distribution Radii cm(in) Temperature, K(F) Power profile fuel-center 0.00000 ( 0.00000) 1348.5 (1967.6) 0.9682 fuel-0.08331 ( 0.03280) 1327.1 (1929.2) 0.9701 fuel-0.15618 ( 0.06149) 1274.6 (1834.6) 0.9750 fuel-0.21931 ( 0.08634) 1205.8 (1710.8) 0.9817 fuel-0.27339 ( 0.10763) 1131.9 (1577.8) 0.9893 fuel-0.31913 ( 0.12564) 1060.4 (1449.1) 0.9970 fuel-0.35724 ( 0.14064) 995.7 (1332.6) 1.0043 fuel-0.38841 ( 0.15292) 939.9 (1232.2) 1.0110 fuel-0.41336 ( 0.16274) 893.9 (1149.3) 1.0168 fuel-0.43278 ( 0.17039) 857.3 (1083.5) 1.0216 fuel-0.44738 ( 0.17613) 829.6 (1033.6) 1.0254 fuel-0.45783 ( 0.18025) 809.6 ( 997.5) 1.0282 fuel-0.46483 ( 0.18300) 796.1 ( 973.3) 1.0301 fuel-0.46908 ( 0.18468) 787.9 ( 958.5) 1.0313 fuel-0.47126 ( 0.18554) 783.7 ( 951.0) 1.0320 fuel-0.47207 ( 0.18585) 782.1 ( 948.2) 1.0324 fuel-outer surface 0.47218 ( 0.18590) 781.9 ( 947.8) 1.0325 cladding inner surface 0.47942 ( 0.18875) 628.0 ( 670.8) cladding outer surface 0.54691 ( 0.21532) 597.9 ( 616.6) oxide surface 0.54692 ( 0.21532) 597.9 ( 616.5) coolant temperature 573.3 ( 572.3)

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51 D.2. FRAPCON-AM/MD BOL Radial Temperatu re Profile (node 4/12): xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x **** FRAPCON-3.3 (Aug. 12 05) **** x x Released August 2005 x x Oconee rod 15309 x x run date: 07-Oct-31 page 8 x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx axial region number 5 power-time step 1 avg. linear heat rating, kW/m(kW/ft) 19.03( 5.80) local linear heat rating, kW/m(kW/ft) 23.70( 7.22) rod surface heat flux, W/m**2(btu/hr-ft**2) 6.90E+05(2.19E+05) peak linear heat rating, kW/m(kW/ft) 23.93( 7.29) step starts at time, days(sec) 0.00( 0.00E+00) starting burnup, MWd/kgU(MWd/mtU) 0.00( 0.) time increment, days(sec) 0.10( 8.64E+03) burnup increment, MWd/kgU(MWd/mtU) 0.00( 4.) end step, days(sec) 0.10( 8.64E+03) end step burnup, MWd/kgU(MWd/mtU) 0.00( 4.) Radial temperature and power distribution Radii cm(in) Temperature, K(F) Power profile fuel-center 0.00000 ( 0.00000) 1364.6 (1996.6) 0.9682 fuel-0.08332 ( 0.03280) 1339.8 (1951.9) 0.9701 fuel-0.15621 ( 0.06150) 1279.7 (1843.8) 0.9750 fuel-0.21934 ( 0.08635) 1203.3 (1706.3) 0.9817 fuel-0.27341 ( 0.10764) 1123.9 (1563.4) 0.9893 fuel-0.31915 ( 0.12565) 1049.7 (1429.7) 0.9970 fuel-0.35725 ( 0.14065) 984.5 (1312.5) 1.0043 fuel-0.38842 ( 0.15292) 929.9 (1214.2) 1.0110 fuel-0.41337 ( 0.16274) 885.9 (1135.0) 1.0168 fuel-0.43279 ( 0.17039) 851.6 (1073.3) 1.0216 fuel-0.44738 ( 0.17613) 826.0 (1027.0) 1.0254 fuel-0.45783 ( 0.18025) 807.6 ( 994.1) 1.0281 fuel-0.46483 ( 0.18301) 795.4 ( 972.1) 1.0301 fuel-0.46908 ( 0.18468) 788.0 ( 958.8) 1.0313 fuel-0.47126 ( 0.18554) 784.2 ( 952.0) 1.0320 fuel-0.47207 ( 0.18585) 782.8 ( 949.4) 1.0324 fuel-outer surface 0.47218 ( 0.18590) 782.6 ( 949.1) 1.0325 cladding inner surface 0.47943 ( 0.18875) 630.6 ( 675.5) cladding outer surface 0.54692 ( 0.21532) 600.9 ( 621.9) oxide surface 0.54693 ( 0.21533) 600.8 ( 621.8) coolant temperature 576.6 ( 578.3)

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52 LIST OF REFERENCES 1Pacific_Northwest_National_Laboratory (1997). FRAPCON-3: A Computer Code for the Calculation of Steady-State, Thermal-mechani cal Behavior of Oxide Fuel Rods. Pacific Northwest National Laboratory. 2 U.S. Department of Energy (2006) The Global Nuclear Ener gy Partnership Statement of Principles. http://www.gnep.energy.gov/ 3 Berna, G. A., Beyer, C.E., Davis, K.L., Lanning, D.D. (1997). FRAPCON-3: A computer Code for the Calculation os Steady-S tate, Thermal-Mechanical Behavior of Oxide Fuel Rods for High Burnup. U.S. Nuclear Regulatory Commission. Washington, D.C., NUREG/CR-6534. 4 Idaho_National_Laborator y (2005). Relap5. Relap-5 Idaho Falls, Idaho National Laboratory: Hydrodynamics and Reactor Kinetics Code. 5 D. D. Lanning, C. E. B., K.J.Geelhood (2005) FRAPCON-3 Updates, Including Mixed-Oxide Fuel Properties. U. S. N. R. Commission. Washington, D.C., Pacific Northwest National Laboratory. vol 4. 6 Ohira, K., Itagaki, N. (1997). Thermal Conduc tivity Measurements of High Burnup UO2 Pellet and a Benchmark Calculation of Fuel Center Temperature ANS International Topical Meeting on LWR Fuel Performance, Portland, Oregon, American Nuclear Society. 7 Lucuta, P. G., H.S. Matzke, and I.J. Hasti ngs (1996). "A Pragmatic Approach to Modeling Thermal Conductivity of Irradiated UO2 Fuel: Review and Recommendations." Journal of Nuclear Materials 232: 166-180. 8 Ronchi, C., M. Sheindlin, M. Musella, and G.J. Hyland. (1999.). "Thermal Conductivity of Uranium Dioxide Up to 2900K from Simultaneous Measurement of the Heat Capacity and Thermal Diffusivity." Journal of Applied Physics 85(2): 776-789. 9 Bohn, M. P. (1977). "FRACAS: A Subcode fo r the Analysis of Fuel Pellet-Cladding Mechanical Interaction." TREE-NUREG-1028 10 NRC. (2004). "U.S. Nuclear Power Plants: Oconee." http://www.eia.doe.gov/cneaf/nuclear/pag e/at_a_glance/reactors/oconee.htm l. 11 Schlick, T. (1996). Pursuing Laplace's Vision on Modern Computers. IMA Volumes in Mathematics and Its Applications New York, Springer-Verlag. 82: 218-247. 12 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1987). 13 D. Frenkel and B. Smit, Molecular Dynamics Simulation 2nd ed. (Academic Press, 2002).

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53 14 T. Watanabe, S. B. Sinnott, J. S. Tulenko, R. W. Grimes, P. K. Schelling, and S. R. Phillpot, J. Nucl. Mater., (in print) (2008). 15 T. Watanabe, S. G. Srivilliputhur, P. K. Sc helling, J. S. Tulenko, S. B. Sinnott and S. R. Phillpot, J. Am. Cer. Soc., (in preparation) (2008). 16 Watanabe, T. (2008). Thermal Transport in Uranium Dioxide and Diamond by Atomic Level Simulations. Department of Materials Science and Engineering Gainesville, University of Florida. Ph.D Dissertation: 269. 17 N. W. Ashcroft and D. N. Mermin, Solid State Physics (Saunders College Publishing, 1976). 18 K. Govers, S. Lemehov, and M. Verwerft, J. Nucl. Mater. 266, 161 (2007).

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54 BIOGRAPHICAL SKETCH Daniel Vega graduated from Franklin High Sc hool in El Paso, TX in 2000, and enrolled at Texas A&M University. While an undergra duate, Daniel studied physics and nuclear engineering, worked as a reactor operator at the Texas A&M Nuclear Science Center, and spent a year working on intern ational nuclear safeguards for the In ternational Atomic Energy Agency (IAEA) in Vienna, Austria. He graduated with a B.S. in nuclear engineering in December 2005, and pursued his maritime interests as a marine assistant on a National Science Foundation (NSF) funded marine geophysics cruise in Antarctica. Then, Mr. Vega enro lled as a graduate student at the University of Florida in the Department of Nuclear and Radiological Engineering, and has recently received an M.S. in nuclear engineerin g. He has accepted a position with the U.S. Department of Energys Office of Nuclear En ergy, and begins work in August 2008. His interests include outdoor activities, travel, music, and his dog, Sandwich.